a note on set theoretic solutions of the yang baxter equation abstract this paper shows that every finite non degenerate involutive set theoretic solution x r of the yang baxter equation whose permutation group g x r has cardinality which is a cube free number is a multipermutation solution some properties of finite braces are also investigated it is also shown that if a is a left brace whose cardinality is an odd number and a b a b for all a b a then a is a two sided brace and hence a jacobson radical ring it is also observed that the semidirect product and the wreath product of braces of a finite multipermutation level is a brace of a finite multipermutation level

yang baxter deformations and rack cohomology abstract in his study of quantum groups drinfeld suggested to consider set theoretic solutions of the yang baxter equation as a discrete analogon as a typical example every conjugacy class in a group or more generally every rack q provides such a yang baxter operator cq x y 7 y x y in this article we study deformations of cq within the space of yang baxter operators infinitesimally these deformations are classified by yang baxter cohomology we show that the yang baxter cochain complex of cq homotopyretracts to a much smaller subcomplex called quasi diagonal this greatly simplifies the deformation theory of cq including the modular case which had previously been left in suspense by establishing that every deformation of cq is gauge equivalent to a quasi diagonal one in a quasi diagonal deformation only behaviourally equivalent elements of q interact if all elements of q are behaviourally distinct then the yang baxter cohomology of cq collapses to its diagonal part which we identify with rack cohomology this establishes a strong relationship between the classical deformation theory following gerstenhaber and the more recent cohomology theory of racks both of which have numerous applications in knot theory

exponential stability of impulsive systems with application to uncertain sampled data systems we establish exponential stability of nonlinear time varying impulsive systems by employing lyapunov functions with discontinuity at the impulse times our stability conditions have the property that when specialized to linear impulsive systems the stability tests can be formulated as linear matrix inequalities lmis then we consider lti uncertain sampled data systems in which there are two sources of uncertainty the values of the process parameters can be unknown while satisfying a polytopic condition and the sampling intervals can be uncertain and variable we model such systems as linear impulsive systems and we apply our theorem to the analysis and state feedback stabilization we find a positive constant which determines an upper bound on the sampling intervals for which the stability of the closed loop is guaranteed the control design lmis also provide controller gains that can be used to stabilize the process we also consider sampled data systems with constant sampling intervals and provide results that are less conservative than the ones obtained for variable sampling intervals

matched pairs approach to set theoretic solutions of the yang baxter equation we study set theoretic solutions x r of the yang baxter equations on a set x in terms of the induced left and right actions of x on itself we give a characterisation of involutive square free solutions in terms of cyclicity conditions we characterise general solutions in terms of abstract matched pair properties of the associated monoid s x r and we show that r extends as a solution rs on s x r as a set finally we study extensions of solutions both directly and in terms of matched pairs of their associated monoids we also prove several general results about matched pairs of monoids s of the required type including iterated products s s s equivalent to rs a solution and extensions s t rs t examples include a general double construction s s rs s and some concrete extensions their actions and graphs based on small sets

optimal approximation of sparse hessians and its equivalence to a graph coloring problem we consider the problem of approximating the hessian matrix of a smooth non linear function using a minimum number of gradient evaluations particularly in the case that the hessian has a known fixed sparsity pattern we study the class of direct methods for this problem and propose two new ways of classifying direct methods examples are given that show the relationships among optimal methods from each class the problem of finding a non overlapping direct cover is shown to be equivalent to a generalized graph coloring problem the distance 2 graph coloring problem a theorem is proved showing that the general distance k graph coloring problem is np complete for all fixedk 2 and hence that the optimal non overlapping direct cover problem is also np complete some worst case bounds on the performance of a simple coloring heuristic are given an appendix proves a well known folklore result which gives lower bounds on the number of gradient evaluations needed in any possible approximation method

classical solutions of the quantum yang baxter equation the classical analogue is developed here for part of the construction in which knot and link invariants are produced from representations of quantum groups whereas previous work begins with a quantum group obtained by deforming the multiplication of functions on a poisson lie group we work directly with a poisson lie groupg and its associated symplectic groupoid the classical analog of the quantumr matrix is a lagrangian submanifold open image in new window in the cartesian square of the symplectic groupoid for any symplectic leafs ing open image in new window induces a symplectic automorphism ofs s which satisfies the set theoretic yang baxter equation when combined with the flip map exchanging components and suitably implanted in each cartesian powers n generates a symplectic action of the braid groupb n ons n application of a symplectic trace formula to the fixed point set of the action of braids should lead to link invariants but work on this last step is still in progress

convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled data systems under dwell time constraints stability analysis and control of linear impulsive systems is addressed in a hybrid framework through the use of continuous time time varying discontinuous lyapunov functions necessary and sufficient conditions for stability of impulsive systems with periodic impulses are first provided in order to set up the main ideas extensions to the stability of aperiodic systems under minimum maximum and ranged dwell times are then derived by exploiting further the particular structure of the stability conditions the results are non conservatively extended to quadratic stability analysis of linear uncertain impulsive systems these stability criteria are in turn losslessly extended to stabilization using a particular yet broad enough class of state feedback controllers providing then a convex solution to the open problem of robust dwell time stabilization of impulsive systems using hybrid stability criteria relying finally on the representability of sampled data systems as impulsive systems the problems of robust stability analysis and robust stabilization of periodic and aperiodic uncertain sampled data systems are straightforwardly solved using the same ideas several examples are discussed in order to show the effectiveness and reduced complexity of the proposed approach

from racks to pointed hopf algebras abstract a fundamental step in the classification of finite dimensional complex pointed hopf algebras is the determination of all finite dimensional nichols algebras of braided vector spaces arising from groups the most important class of braided vector spaces arising from groups is the class of braided vector spaces c x c q where x is a rack and q is a 2 cocycle on x with values in c racks and cohomology of racks appeared also in the work of topologists this leads us to the study of the structure of racks their cohomology groups and the corresponding nichols algebras we will show advances in these three directions we classify simple racks in group theoretical terms we describe projections of racks in terms of general cocycles we introduce a general cohomology theory of racks containing properly the existing ones we introduce a fourier transform on racks of certain type finally we compute some new examples of finite dimensional nichols algebras

right l groups geometric garside groups and solutions of the quantum yang baxter equation abstract right lattice ordered groups are introduced and studied as a general framework for garside groups and related groups without a garside element every right l group g has a two sided partially ordered subgroup n g which generalizes the quasi centre of an artin tits group the group n g splits into copies of z if g is noetherian the positive cone of a right l group is described as a structure that is known from algebraic logic a pair of left and right self similar hoops for noetherian right l groups g modularity of the lattice structure is characterized in terms of an operation on the set x g of atoms it is proved that modular garside groups are equivalent to finite projective spaces with a non degenerate labelling a concept of duality for x g is introduced and applied in the distributive case this gives a one to one correspondence between noetherian right l groups with duality and non degenerate unitary set theoretic solutions of the quantum yang baxter equation the description of garside groups via garside germs is extended to right l groups which yields a one sided enhancement and a new proof of dvurecenskij s non commutative extension of mundici s correspondence between abelian l groups and mv algebras

an extension of lasalle s invariance principle for switched systems in this paper we address invariance principles for a certain class of switched nonlinear systems we provide an extension of lasalle s invariance principle for these systems and state asymptotic stability criteria we also present some related results on the compactness of the trajectories of these switched systems

a scalable parallel algorithm for incomplete factor preconditioning we describe a parallel algorithm for computing incomplete factor ilu preconditioners the algorithm attains a high degree of parallelism through graph partitioning and a two level ordering strategy both the subdomains and the nodes within each subdomain are ordered to preserve concurrency we show through an algorithmic analysis and through computational results that this algorithm is scalable experimental results include timings on three parallel platforms for problems with up to 20 million unknowns running on up to 216 processors the resulting preconditioned krylov solvers have the desirable property that the number of iterations required for convergence is insensitive to the number of processors

a family of solutions of the yang baxter equation a new method to construct involutive non degenerate set theoretic solutions xn r n of the yang baxter equation for any positive integer n from a given solution x r is presented furthermore the permutation group g xn r n associated with the solution xn r n is isomorphic to a subgroup of g x r and in many cases g xn r n g x r

on the set theoretical yang baxter equation we propose a general way of constructing set theoretical solutions of the yangbaxter equation we study the properties of the construction we also show that our construction includes the earlier ones given by weinstein xu and etingof schedlersoloviev

affine structures of decomposable solvable groups abstract bi crossed products of groups are closely related to braces a ring like structure connected with solutions to the yang baxter equation the asymmetric product of braces which has become useful for the construction of regular affine groups and simple braces as well as several other product constructions are unified extended and simplified all bi crossed products of bicyclic braces are classified

multipermutation solutions of the yang baxter equation set theoretic solutions of the yang baxter equation form a meeting ground of mathematical physics algebra and combinatorics such a solution consists of a set x and a function r x x x x which satisfies the braid relation

ilum a multi elimination ilu preconditioner for general sparse matrices standard preconditioning techniques based on incomplete lu ilu factorizations offer a limited degree of parallelism in general a few of the alternatives advocated so far consist of either using some form of polynomial preconditioning or applying the usual ilu factorization to a matrix obtained from a multicolor ordering in this paper we present an incomplete factorization technique based on independent set orderings and multicoloring we note that in order to improve robustness it is necessary to allow the preconditioner to have an arbitrarily high accuracy as is done with ilus based on threshold techniques the ilum factorization described in this paper is in this category it can be viewed as a multifrontal version of a gaussian elimination procedure with threshold dropping which has a high degree of potential parallelism the emphasis is on methods that deal specifically with general unstructured sparse matrices such as those arising from finite element methods on unstructured meshes

a covering theory for non involutive set theoretic solutions to the yang baxter equation abstract an algebraic theory of non involutive set theoretic solutions to the yang baxter equation by means of a non commutative version of braces is established the structure group and the permutation group of a solution is put into a context of discrete lie theory this leads to a natural concept of covering which shares the good properties of coverings in geometry and topology in the special case of quandle coverings in knot theory known anomalies are removed

on groups of i type and involutive yang baxter groups abstract we suggest a cohomological framework to describe groups of i type and involutive yang baxter groups these groups are key in the study of involutive non degenerate set theoretic solutions of the quantum yang baxter equation our main tool is a lifting criterion for 1 cocycles established here in a general non abelian setting

self inverse yang baxter operators from co algebra structures some attempts to find yang baxter operators in a systematic way have led to the theory of quantum groups see 4 many papers in the literature are devoted to the construction of them see 5 p 198 we adopt the terminology of 5 in giving some relevant examples the twist operators on modules over braided bialgebras and operators on comodules over cobraided bialgebras the frt construction showed that any yang baxter operator on a finite dimensional space v can be obtained from a cobraided bialgebra coacting on v other authors might use the term quasi triangular bialgebras instead of braided bialgebras see for example 4 7 all these constructions have proved of great utility in low dimensional topology we present here a new method to construct self inverse yang baxter operators thus to every co algebra structure on a space we associate an operator we describe the algebra respective coalgebra structures producing the same operator yang baxter operators associated to algebras are different from those associated to coalgebras except for a special case we give characterizations for operators which are associated to co algebra structures and construct the structures which produce them we give an

frt construction for dynamical yang baxter maps abstract notions of an h x bialgebroid and of its dynamical representation are proposed the dynamical representations of each h x bialgebroid form a tensor category every dynamical yang baxter map r satisfying suitable conditions a generalization of the set theoretical solution to the quantum yang baxter equation gives birth to an h x bialgebroid a r the categories of l operators for r and of dynamical representations of a r are isomorphic as tensor categories

a decomposition theorem for square free unitary solutions of the quantum yang baxter equation abstract it is known that every skew polynomial ring with generating set x and binomial relations in the sense of gateva ivanova trans amer math soc 343 1994 203 is an artin schelter regular domain of global dimension x moreover every such ring gives rise to a non degenerate unitary set theoretical solution r x 2 x 2 of the quantum yang baxter equation which fixes the diagonal of x 2 gateva ivanova s conjecture talk at the international algebra conference miskolc hungary 1996 states that conversely every such solution r comes from a skew polynomial ring with binomial relations an equivalent conjecture duke math j 100 1999 169 says that the underlying set x is r decomposable we prove these conjectures and construct an indecomposable solution r with x which shows that an extension to infinite x is false

poisson yang baxter maps with binomial lax matrices a construction of multidimensional parametric yang baxter maps is presented the corresponding lax matrices are the symplectic leaves of first degree matrix polynomials equipped with the sklyanin bracket these maps are symplectic with respect to the reduced symplectic structure on these leaves and provide examples of integrable mappings an interesting family of quadrirational symplectic yb maps on c4 c4 with 3 3 lax matrices is also presented

estimation of sparse jacobian matrices and graph coloring blems given a mapping with a sparse jacobian matrix the problem of minimizing the number of function evaluations needed to estimate the jacobian matrix by differences is investigated this problem can be attacked as a graph coloring problem and this approach leads to very efficient algorithms the behavior of these algorithms is studied and in particular it is proved that two of the algorithms are optimal for band graphs numerical evidence is presented which indicates that these two algorithms are nearly optimal on practical problems

the interplay between k graphs and the yang baxter equation in this paper we initiate the study of the interplay between k graphs and the yang baxter equation for this we provide two very different perspectives on one hand we show that the set of all set theoretic solutions of the yang baxter equation is a special class of single vertex k graphs as a consequence we construct an infinite family of large solutions of the yang baxter equation from an arbitrarily given one on the other hand we prove that all single vertex k graphs are yb semigroups of square free involutive solutions of the yang baxter equation other various connections are also investigated

a combinatorial approach to the set theoretic solutions of the yang baxter equation a bijective map r x2 x2 where x x1 xn is a finite set is called a set theoretic solution of the yang baxter equation ybe if the braid relation r12r23r12 r23r12r23 holds in x3 a nondegenerate involutive solution x r satisfying r xx xx for all x x is called square free solution there exist close relations between the square free set theoretic solutions of ybe the semigroups of i type the semigroups of skew polynomial type and the bieberbach groups as it was first shown in a joint paper with michel van den bergh in this paper we continue the study of square free solutions x r and the associated yang baxter algebraic structures the semigroup s x r the group g x r and the k algebra a k x r over a field k generated by x and with quadratic defining relations naturally arising and uniquely determined by r we study the properties of the associated yang baxter structures and prove a conjecture of the present author that the three notions a square free solution of set theoretic ybe

semi braces and the yang baxter equation abstract in this paper we obtain new solutions of the yang baxter equation that are left non degenerate through left semi braces a generalization of braces introduced by rump in order to provide new solutions we introduce the asymmetric product of left semi braces a generalization of the semidirect product of braces that allows us to produce several examples of left semi braces

yang baxter maps with first degree polynomial 2 2 lax matrices a family of non parametric yang baxter yb maps is constructed by re factorization of 2 2 matrix polynomials of first degree these maps are poisson with respect to the sklyanin bracket for each casimir function a parametric poisson yb map is generated by reduction on the corresponding level set by considering a complete set of casimir functions symplectic multi parametric yb maps are derived these maps are quadrirational with explicit formulae in terms of matrix operations their lax matrices are by construction 2 2 first degree polynomial in the spectral parameter and are classified by the jordan normal form of the leading term degenerate parametric yb maps constructed as limits of the non degenerate ones are connected to known integrable systems on quadgraphs

a converse lyapunov theorem for linear parameter varying and linear switching systems we study families of linear time varying systems where time variations have to satisfy restrictions on the dwell time that is on the minimum distance between discontinuities as well as on the derivative in between discontinuities such classes of systems may be formulated as linear flows on vector bundles the main objective of this paper is to construct parameter dependent lyapunov functions which characterize the exponential growth rate this is possible in the generic irreducible case as an application the gelfand formula is generalized to the class of systems studied here in other words the maximal exponential growth rate may be approximated by only considering the periodic systems in the family of time varying systems a perspective on the question of continuous dependence of the exponential growth rate on the data is given

quadratic algebras yang baxter equation and artin schelter regularity abstract we study two classes of quadratic algebras over a field k the class c n of all n generated pbw algebras with polynomial growth and finite global dimension and the class of quantum binomial algebras we show that a pbw algebra a is in c n iff its hilbert series is h a z 1 1 z n furthermore the class c n contains a unique up to isomorphism monomial algebra a k x 1 x n x j x i 1 i j n a surprising amount can be said when a is a quantum binomial algebra that is its defining relations are nondegenerate square free binomials x y c x y z t c x y k our main result shows that for an n generated quantum binomial algebra a the following conditions are equivalent i a is an artin schelter regular pbw algebra ii a is a yang baxter algebra that is the set of relations defines canonically a solution of the yang baxter equation iii a is a binomial skew polynomial ring with respect to an enumeration of x iv the koszul dual a is a quantum grassmann algebra

stability and stabilization of continuous time switched linear systems this paper addresses two strategies for the stabilization of continuous time switched linear systems the first one is of open loop nature trajectory independent and is based on the determination of a minimum dwell time by means of a family of quadratic lyapunov functions the relevant point on dwell time calculation is that the proposed stability condition does not require the lyapunov function to be uniformly decreasing at every switching time the second one is of closed loop nature trajectory dependent and is designed from the solution of what we call lyapunov metzler inequalities from which the stability condition including chattering is expressed being nonconvex a more conservative but simpler to solve version of the lyapunov metzler inequalities is provided the theoretical results are illustrated by means of examples

skew lattices and set theoretic solutions of the yang baxter equation abstract in this paper we discuss and characterize several set theoretic solutions of the yang baxter equation obtained using skew lattices an algebraic structure that has not yet been related to the yang baxter equation such solutions are degenerate in general and thus different from solutions obtained from braces and other algebraic structures our main result concerns a description of a set theoretic solution of the yang baxter equation obtained from an arbitrary skew lattice we also provide a construction of a cancellative and distributive skew lattice on a given family of pairwise disjoint sets

solutions of the yang baxter equation associated with a left brace abstract given a left brace b a method is given to construct explicitly all the non degenerate involutive set theoretic solutions x r of the yang baxter equation such that the associated permutation group g x r is isomorphic as a left brace to b this method depends entirely on the brace structure of b

on hopf algebras with positive bases we show that if a finite dimensional hopf algebra h over c has a basis with respect to which all the structure constants are nonnegative then h is isomorphic to the bi cross product hopf algebra constructed by takeuchi and majid from a finite group g and a unique factorization g g g of g into two subgroups we also show that hopf algebras in the category of finite sets with correspondences as morphisms are classified in a similar way our results can be used to explain some results on hopf algebras from the set theoretical point of view

stability and robustness analysis for switched systems with time varying delays a new technique is presented for the stability and robustness analysis of nonlinear switched time varying systems with uncertainties and time varying delays the delays are allowed to be discontinuous but are required to be piecewise continuous and arbitrarily long with known upper bounds the technique uses an adaptation of halanay s inequality and a trajectory based technique and is used for designing switched controllers to stabilize linear time varying systems with time varying delays

braces and the yang baxter equation several aspects of relations between braces and non degenerate involutive set theoretic solutions of the yang baxter equation are discussed and many consequences are derived in particular for each positive integer n a finite square free multipermutation solution of the yang baxter equation with multipermutation level n and an abelian involutive yang baxter group is constructed this answers a problem of gateva ivanova and cameron it is proved that finite non degenerate involutive set theoretic solutions of the yang baxter equation whose associated involutive yang baxter group is abelian are multipermutation solutions earlier the authors proved this with the additional square free hypothesis on the solutions it is also proved that finite square free non degenerate involutive set theoretic solutions associated to a left brace are multipermutation solutions

uniform asymptotic stability of nonlinear switched systems with an application to mobile robots this paper is concerned with the study of both local and global uniform asymptotic stability for general nonlinear and time varying switched systems two concepts of lyapunov functions are introduced and used to establish uniform lyapunov stability and uniform global stability with the help of output functions an almost bounded output energy condition and an output persistent excitation condition are then proposed and employed to guarantee uniform local and global asymptotic stability based on this result a generalized version of krasovskii lasalle theorem in time varying switched systems is proposed for switched systems with persistent dwell time the output persistent excitation condition is guaranteed to hold under a zero state observability condition it is shown that several existing results in past literature can be covered as special cases using the proposed criteria interestingly as opposed to previous work the main results of this paper are applicable to the situation where some switching systems are not asymptotically stable at the origin the robust practical regulation problem of nonholonomic mobile robots is studied as a way of demonstrating the power of the proposed new criteria a novel switching controller is proposed with guaranteed robustness to orientation error and unknown parameters in mobile robots

combinatorial solutions to the reflection equation abstract we use ring theoretic methods and methods from the theory of skew braces to produce set theoretic solutions to the reflection equation we also use set theoretic solutions to construct solutions of the parameter dependent reflection equation

on converse lyapunov theorems for iss and iiss switched nonlinear systems abstract in this paper we present converse lyapunov theorems for iss and integral input to state stable iiss switched nonlinear systems their proofs are based on existing converse lyapunov theorems for input output to state stable and iiss nonlinear systems and on the association of the switched system with a nonlinear system with inputs and disturbances that take values in a compact set

involutive yang baxter groups in 1992 drinfeld posed the question of finding the set theoretic solutions of the yang baxter equation recently gateva ivanova and van den bergh and etingof schedler and soloviev have shown a group theoretical interpretation of involutive non degenerate solutions namely there is a one to one correspondence between involutive non degenerate solutions on finite sets and groups of i type a group g of i type is a group isomorphic to a subgroup of fa n sym n so that the projection onto the first component is a bijective map where fa n is the free abelian group of rank n and sym n is the symmetric group of degree n the projection of g onto the second component sym n we call an involutive yang baxter group iyb group this suggests the following strategy to attack drinfeld s problem for involutive non degenerate set theoretic solutions first classify the iyb groups and second for a given iyb group g classify the groups of i type with g as associated iyb group it is known that every iyb group is solvable in this paper some results supporting the converse of this property are obtained more precisely we show that some classes of groups are iyb groups we also give a non obvious method to construct infinitely many groups of i type and hence infinitely many involutive non degenerate set theoretic solutions of the yang baxter equation with a prescribed associated iyb group

distributed memory parallel algorithms for distance 2 coloring and related problems in derivative computation the distance 2 graph coloring problem aims at partitioning the vertex set of a graph into the fewest sets consisting of vertices pairwise at distance greater than 2 from each other its applications include derivative computation in numerical optimization and channel assignment in radio networks we present efficient distributed memory parallel heuristic algorithms for this np hard problem as well as for two related problems used in the computation of jacobians and hessians parallel speedup is achieved through graph partitioning speculative iterative coloring and a bulk synchronous parallel like organization of parallel computation results from experiments conducted on a pc cluster employing up to 96 processors and using large size real world as well as synthetically generated test graphs show that the algorithms are scalable in terms of quality of solution the algorithms perform remarkably well the numbers of colors used by the parallel algorithms are observed to be very close to the numbers used by their sequential counterparts which in turn are quite often near optimal moreover the experimental results show that the parallel distance 2 coloring algorithm compares favorably with the alternative approach of solving the distance 2 coloring problem on a graph mathcal g by first constructing the square graph mathcal g 2 and then applying a parallel distance 1 coloring algorithm on mathcal g 2 implementations of the algorithms are made available via the zoltan toolkit

new acyclic and star coloring algorithms with application to computing hessians acyclic and star coloring problems are specialized vertex coloring problems that arise in the efficient computation of hessians using automatic differentiation or finite differencing when both sparsity and symmetry are exploited we present an algorithmic paradigm for finding heuristic solutions for these two np hard problems the underlying common technique is the exploitation of the structure of two colored induced subgraphs for a graph g on n vertices and m edges the time complexity of our star coloring algorithm is o n overline d 2 where overline d k a generalization of vertex degree denotes the average number of distinct paths of length at most k edges starting at a vertex in g the time complexity of our acyclic coloring algorithm is larger by a multiplicative factor involving the inverse of ackermann s function the space complexity of both algorithms is o m to the best of our knowledge our work is the first practical algorithm for the acyclic coloring problem for the star coloring problem our algorithm uses fewer colors and is considerably faster than a previously known o n overline d 3 time algorithm computational results from experiments on various large size test graphs demonstrate that the algorithms are fast and produce highly effective solutions the use of these algorithms in hessian computation is expected to reduce overall runtime drastically

periodical stabilization of switched linear systems periodical stabilization problems for switched linear systems are investigated in this paper for autonomous switched systems if there exists a stable convex combination of the subsystems then a periodically switching signal can be constructed such that the overall system is asymptotically stable based on this fact for switched control systems corresponding sufficient conditions are presented under which constant switching direct observer based state feedback controller can be designed such that the corresponding closed loop systems are asymptotically stable under some periodically switching signal some numerical examples are given to illustrate our results

set theoretic solutions of the yang baxter equation braces and symmetric groups abstract we involve simultaneously the theory of braided groups and the theory of braces to study set theoretic solutions of the yang baxter equation ybe we show the intimate relation between the notions of a symmetric group in the sense of takeuchi i e a braided involutive group and a left brace we find new results on symmetric groups of finite multipermutation level and the corresponding braces we introduce a new invariant of a symmetric group g r the derived chain of ideals of g which gives a precise information about the recursive process of retraction of g we prove that every symmetric group g r of finite multipermutation level m is a solvable group of solvable length m to each set theoretic solution x r of ybe we associate two invariant sequences of involutive braided groups i the sequence of its derived symmetric groups ii the sequence of its derived permutation groups and explore these for explicit descriptions of the recursive process of retraction of x r we find new criteria necessary and sufficient to claim that x r is a multipermutation solution

convex conditions for robust stabilization of uncertain switched systems with guaranteed minimum and mode dependent dwell time abstract alternative conditions for establishing dwell time stability properties of linear switched systems are considered unlike the hybrid conditions derived in geromel and colaneri 2006 the considered ones are affine in the system matrices allowing then for the consideration of uncertain switched systems with time varying uncertainties the low number of decision variables moreover permits to easily derive convex stabilization conditions using a specific class of state feedback control laws the resulting conditions are enforced using sum of squares programming which are shown to be less complex numerically that approaches based on piecewise linear functions or looped functionals previously considered in the literature the sums of squares conditions are also proven to 1 approximate arbitrarily well the conditions of geromel and colaneri 2006 and 2 be invariant with respect to time scaling emphasizing that the complexity of the approach does not depend on the size of the dwell time several comparative examples illustrate the efficiency of the approach

estimation of sparse hessian matrices and graph coloring problems large scale optimization problems often require an approximation to the hessian matrix if the hessian matrix is sparse then estimation by differences of gradients is attractive because the number of required differences is usually small compared to the dimension of the problem the problem of estimating hessian matrices by diferences can be phrased as follows given the sparsity structure of a symmetric matrix a obtain vectors d 1 d 2 ldots d p such that ad 1 ad 2 ldots ad p determine a uniquely with p as small as possible we approach this problem from a graph theoretic point of view and show that both direct and indirect approaches to this problem have a natural graph coloring interpretation the complexity of the problem is analyzed and efficient practical heuristic procedures are developed numerical results illustrate the differences between the various approaches

homology of left non degenerate set theoretic solutions to the yang baxter equation abstract this paper deals with left non degenerate set theoretic solutions to the yang baxter equation lnd solutions a vast class of algebraic structures encompassing groups racks and cycle sets to each such solution there is associated a shelf i e a self distributive structure which captures its major properties we consider two co homology theories for lnd solutions one of which was previously known in a reduced form for biracks only an explicit isomorphism between these theories is described for groups and racks we recover their classical co homology whereas for cycle sets we get new constructions for a certain type of lnd solutions including quandles and non degenerate cycle sets the co homologies split into the degenerate and the normalized parts we express 2 cocycles of our theories in terms of group cohomology and in the case of cycle sets establish connexions with extensions this leads to a construction of cycle sets with interesting properties

a nonconservative lmi condition for stability of switched systems with guaranteed dwell time ensuring stability of switched linear systems with a guaranteed dwell time is an important problem in control systems several methods have been proposed in the literature to address this problem but unfortunately they provide sufficient conditions only this technical note proposes the use of homogeneous polynomial lyapunov functions in the non restrictive case where all the subsystems are hurwitz showing that a sufficient condition can be provided in terms of an lmi feasibility test by exploiting a key representation of polynomials several properties are proved for this condition in particular that it is also necessary for a sufficiently large degree of these functions as a result the proposed condition provides a sequence of upper bounds of the minimum dwell time that approximate it arbitrarily well some examples illustrate the proposed approach

affine characterizations of minimal and mode dependent dwell times for uncertain linear switched systems an alternative approach for minimum and mode dependent dwell time characterization for switched systems is derived while minimum dwell time results require the subsystems to be asymptotically stable mode dependent dwell time results can consider unstable subsystems and dwell times within a possibly unbounded range of values the proposed approach is related to lyapunov looped functionals a new type of functionals leading to stability conditions affine in the system matrices unlike standard results for minimum dwell time these conditions are expressed as infinite dimensional lmis which can be solved using recent polynomial optimization techniques such as sum of squares the specific structure of the conditions is finally utilized in order to derive dwell time stability results for uncertain switched systems several examples illustrate the efficiency of the approach

on braided groupoids abstract we study and give examples of braided groupoids and a fortiori non degenerate solutions of the quiver theoretical braid equation

one sided orthogonality orthomodular spaces quantum sets and a class of garside groups abstract the classification of orthomodular lattices by means of their structure group is extended to non symmetric orthogonality relations spaces with a non hermitian sesquilinear form quantum sets and a new class of garside groups are given as examples

lyapunov conditions for input to state stability of impulsive systems this paper introduces appropriate concepts of input to state stability iss and integral iss for impulsive systems i e dynamical systems that evolve according to ordinary differential equations most of the time but occasionally exhibit discontinuities or impulses we provide a set of lyapunov based sufficient conditions for establishing these iss properties when the continuous dynamics are iss but the discrete dynamics that govern the impulses are not the impulses should not occur too frequently which is formalized in terms of an average dwell time adt condition conversely when the impulse dynamics are iss but the continuous dynamics are not there must not be overly long intervals between impulses which is formalized in terms of a novel reverse adt condition we also investigate the cases where i both the continuous and discrete dynamics are iss and ii one of these is iss and the other only marginally stable for the zero input while sharing a common lyapunov function in the former case we obtain a stronger notion of iss for which a necessary and sufficient lyapunov characterization is available the use of the tools developed herein is illustrated through examples from a micro electro mechanical system mems oscillator and a problem of remote estimation over a communication network

what color is your jacobian graph coloring for computing derivatives graph coloring has been employed since the 1980s to efficiently compute sparse jacobian and hessian matrices using either finite differences or automatic differentiation several coloring problems occur in this context depending on whether the matrix is a jacobian or a hessian and on the specifics of the computational techniques employed we consider eight variant vertex coloring problems here this article begins with a gentle introduction to the problem of computing a sparse jacobian followed by an overview of the historical development of the research area then we present a unifying framework for the graph models of the variant matrix estimation problems the framework is based upon the viewpoint that a partition of a matrix into structurally orthogonal groups of columns corresponds to distance 2 coloring an appropriate graph representation the unified framework helps integrate earlier work and leads to fresh insights enables the design of more efficient algorithms for many problems leads to new algorithms for others and eases the task of building graph models for new problems we report computational results on two of the coloring problems to support our claims most of the methods for these problems treat a column or a row of a matrix as an atomic entity and partition the columns or rows or both a brief review of methods that do not fit these criteria is provided we also discuss results in discrete mathematics and theoretical computer science that intersect with the topics considered here

braces radical rings and the quantum yang baxter equation abstract non degenerate cycle sets are equivalent to non degenerate unitary set theoretical solutions of the quantum yang baxter equation we embed such cycle sets into generalized radical rings braces and study their interaction in this context we establish a galois theory between ideals of braces and quotient cycle sets our main result determines the relationship between two square free cycle sets operating transitively on each other