primitive zero symmetric sign pattern matrices with the maximum base abstract in b cheng b liu the base sets of primitive zero symmetric sign pattern matrices linear algebra appl 428 2008 715 731 cheng and liu studied the bases of primitive zero symmetric sign pattern matrices the sharp upper bound of the bases was obtained in this paper we characterize the sign pattern matrices with the sharp bound

bounds on the local bases of primitive non powerful minimally strong signed digraphs abstract in this paper we study the local bases of primitive non powerful minimally strong signed digraphs of order n 7 we obtain the first two or three largest kth local bases depending on whether n is odd or even together with complete characterization of the equality cases for primitive non powerful minimally strong signed digraphs

bounds on the bases of irreducible generalized sign pattern matrices abstract li et al z li f hall c eschenbach on the period and base of a sign pattern matrix linear algebra appl 212 213 1994 101 120 extended the concept of the base or index of convergence and period from nonnegative matrices to powerful sign pattern matrices in this paper we study the bases for non powerful irreducible sign pattern matrices and more generally for generalized sign pattern matrices we obtain sharp upper bounds together with a complete characterization of the equality cases of the bases for both primitive and irreducible sign pattern and generalized sign pattern matrices we also show that there exist gaps in the base set of the classes of such matrices

on the index of convergence of an irreducible boolean matrix abstract first we give a new proof of the sharp upper bound for the indices of convergence of n x n irreducible boolean or nonnegative matrices with period p then we characterize the matrices with the largest index

on the exponent of a primitive digraph abstract we characterize the equality case of the upper bound d n s n 2 for the exponent of a primitive digraph in the case s 2 where n is the number of the vertices of the digraph d and s is the length of the shortest elementary circuit of d we also answer a question about the equality case when s 1 the exponent of an n n primitive nonnegative matrix a is the same as the exponent of the associated digraph d a of a so every theorem in this paper can be translated into a theorem about nonnegative matrices

the proof of a conjecture on the lewin number of primitive non powerful signed digraphs abstract suppose s is a primitive non powerful signed digraph a pair of sssd walks of s are two directed walks of s which have the same initial vertex same terminal vertex and same length but different signs the lewin number of s denoted l s is the least positive integer k such that there are both sssd walks of lengths k and k 1 from some vertex u to some vertex v possibly u again of s this paper presents a proof of a conjecture on l s which was put forward by you et al l h you m h liu b l liu bounds on the lewin number for primitive non powerful signed digraphs acta math appl sinica 35 2012 396 407

the base sets of quasi primitive zero symmetric sign pattern matrices with zero trace abstract cheng and liu bo cheng bolian liu the base sets of primitive zero symmetric sign pattern matrices linear algebra appl 428 2008 715 731 showed that the base set of quasi primitive zero symmetric generalized sign pattern matrices is 1 2 2 n the matrices with zero trace play a prominent role in matrix theory in this paper we investigate the bases of quasi primitive zero symmetric generalized sign pattern matrices with zero trace and prove that the base set of such matrices is 2 3 2 n 1

observer based control of systems with slope restricted nonlinearities an observer design is presented which makes use of bounds on the slope of system nonlinearities necessary and sufficient conditions are derived for the feasibility of the design a class of state feedback control laws are characterized which when implemented with the observer states ensure global asymptotic stability one such certainty equivalence design is illustrated on an active magnetic bearing example

irreducible pattern k potent ray pattern matrices abstract a ray pattern is the class of all complex matrices with a specified zero nonzero pattern such that the argument of each nonzero entry is specified the ray pattern a is called pattern k potent if k is the smallest positive integer for which ak 1 a as ray patterns we characterize the irreducible ray patterns that are pattern k potent and we provide a canonical form for patterns with respect to permutation similarity and signature similarity by diagonal unitary matrices

on some universal algebras associated to the category of lie bialgebras in our previous work math 0008128 we studied the set quant k of all universal quantization functors of lie bialgebras over a field k of characteristic zero compatible with the operations of taking duals and doubles we showed that quant k is canonically isomorphic to a product g0 k k where g0 k is a universal group and k is a quotient set of a set b k of families of lie polynomials by the action of a group g k we prove here that g0 k is equal to the multiplicative group 1 k so quant k is as close as it can be to k we also prove that the only universal derivations of lie bialgebras are multiples of the composition of the bracket with the cobracket finally we prove that the stabilizer of any element of b k is reduced to the 1 parameter subgroup generated by the corresponding square of the antipode

primitive non powerful symmetric loop free signed digraphs with given base and minimum number of arcs abstract in b m kim b c song w hwang primitive graphs with given exponents and minimum number of edges linear algebra appl 420 2007 648 662 the minimum number of edges of a simple graph on n vertices with exponent k was determined in this paper we completely determine the minimum number h n k of arcs of primitive non powerful symmetric loop free signed digraphs on n vertices with base k characterize the underlying digraphs which have h n k arcs when k is 2 nearly characterize the case when k is 3 and propose an open problem

uniqueness of coxeter structures on kac moody algebras abstract let g be a symmetrizable kac moody algebra and u g the corresponding quantum group we showed in 1 2 that the braided coxeter structure on integrable category o representations of u g which underlies the r matrix actions arising from the levi subalgebras of u g and the quantum weyl group action of the generalized braid group b g can be transferred to integrable category o representations of g we prove in this paper that up to unique equivalence there is a unique such structure on the latter category with prescribed restriction functors r matrices and local monodromies this extends simplifies and strengthens a similar result of the second author valid when g is semisimple and is used in 3 to describe the monodromy of the rational casimir connection of g in terms of the quantum weyl group operators of u g our main tool is a refinement of enriquez s universal algebras which is adapted to the prop describing a lie bialgebra graded by the non negative roots of g

quantum groups and deformation quantization explicit approaches and implicit aspects deformation quantization which gives a development of quantum mechanics independent of the operator algebra formulation and quantum groups which arose from the inverse scattering method and a study of yang baxter equations share a common idea abstracted earlier in algebraic deformation theory that algebraic objects have infinitesimal deformations which may point in the direction of certain continuous global deformations i e quantizations in deformation quantization the algebraic object is the algebra of observables functions on symplectic phase space whose infinitesimal deformation is the poisson bracket and global deformation a star product in quantum groups it is a hopf algebra generally either of functions on a lie group or often its dual in the topological vector space sense as we briefly explain a completed universal enveloping algebra of a lie algebra with for infinitesimal a matrix satisfying the modified classical yang baxter equation mcybe frequently existence proofs are known but explicit formulas useful for physical applications have been difficult to extract one success here comes from universal deformation formulas udfs expressions built from a lie algebra which deform any algebra on which the lie algebra operates as derivations the most famous of these is the moyal product a special case of a class in which the lie algebra is abelian another comes from recognition that the belavin drinfel d solutions to the mcybe are in fact infinitesimal deformations for which in the case of the special linear groups it is possible to give explicit formulas for the corresponding quantum yang baxter equations this review paper discusses necessarily in brief these and related topics including twisting as a form of udf and finding formulas for preferred deformations of hopf algebras in which the multiplication or comultiplication is rigid and must be preserved in the course of deformation

groups and lie algebras corresponding to the yang baxter equations for a positive integer n we introduce quadratic lie algebras tr n qtr n and finitely discrete groups tr n qtr n naturally associated with the classical and quantum yang baxter equation respectively we prove that the universal enveloping algebras of the lie algebras tr n qtr n are koszul and compute their hilbert series we also compute the cohomology rings for these lie algebras which by koszulity are the quadratic duals of the enveloping algebras finally we construct a basis of u tr n we construct cell complexes which are classifying spaces of the groups tr n and qtr n and show that the boundary maps in them are zero which allows us to compute the integral cohomology of these groups we show that the lie algebras tr n qtr n map onto the associated graded algebras of the malcev lie algebras of the groups tr n qtr n respectively in the case of tr n we use quantization theory of lie bialgebras to show that this map is actually an isomorphism at the same time we show that the groups tr n and qtr n are not formal for n 4

quantization of γ lie bialgebras abstract we introduce the notion of lie bialgebras where is a group these objects give rise to cocommutative co poisson bialgebras for which we construct quantization functors this enlarges the class of co poisson algebras for which a quantization is known our result relies on our earlier work where we showed that twists of lie bialgebras can be quantized we complement this work by studying the behavior of this quantization under compositions of twists

brief growth rate conditions for uniform asymptotic stability of cascaded time varying systems in the last decade several results on cascaded autonomous systems have appeared in the literature the sufficient conditions for the stability and stabilizability of these systems are often related to certain growth rate conditions on the functions which define the dynamics of the system in this paper we analyze three complementary classes of nonlinear time varying systems according to these growth rates for each case we give further results to guarantee uniform global asymptotic stability of the cascade and relate our contributions to input to state stability and other growth rate conditions previously reported the advantage of this type of analysis is that it is often easier to verify such conditions than to find a lyapunov function with a negative definite derivative

a cohomological construction of quantization functors of lie bialgebras abstract we propose a variant to the etingof kazhdan construction of quantization functors we construct the twistor j associated to an associator using cohomological techniques we then introduce a criterion ensuring that the left hopf algebra of a quasitriangular que algebra is flat we prove that this criterion is satisfied at the universal level this gives a construction of quantization functors equivalent to the etingof kazhdan construction

irreducible powerful ray pattern matrices a ray pattern is a matrix each of whose entries is either 0 or a ray in the complex plane originating from 0 but not including 0 a ray pattern is a natural generalization of the concept of a sign pattern whose entries are from the set 0 powers of sign patterns and ray patterns especially patterns whose powers are periodic have been studied in several recent papers a ray pattern a is said to be powerful if ak is unambiguously defined for all positive integers k irreducible powerful ray patterns have been characterized recently in this paper reducible powerful ray patterns are investigated in particular for a powerful ray pattern in frobenius normal form it is shown that the existence of a nonzero entry in an off diagonal block implies that the corresponding irreducible components are related in a certain way further the structure of each of the off diagonal blocks is characterized

sufficient conditions on observability grammian for synchronization in arrays of coupled linear time varying systems synchronizability of stable output coupled identical linear time varying systems is studied it is shown that if the observability grammian satisfies a persistence of excitation condition then there exists a bounded linear time varying feedback law that yields exponential synchronization for all fixed asymmetrical interconnections with connected graphs also a weaker condition on the grammian is given for asymptotic synchronization no assumption is made on the strength of coupling between the systems

bases of primitive nonpowerful sign patterns for a square primitive nonpowerful sign pattern a the base of a denoted by l a is the least positive integer l such that every entry of a l is for a square sign pattern matrix a with order n the associated digraph of a denoted by d a has vertex set v 1 2 n and arc set e i j a i j 0 the associated signed digraph of a denoted by s a is obtained from d a by assigning sign of a i j to arc i j for all i and j in this paper we consider the base set of the primitive nonpowerful sign pattern matrices for a square primitive nonpowerful sign pattern a with order n and base at least 32n 2 2n 4 some properties about the cycles in s a are obtained and a bound on the base is given some sign pattern matrices with given bases are characterized and some gaps in the base set are shown as well

on persistently exciting observers and a non linear separation principle application to the stabilization of a generator we provide a separation principle for a class of non linear time varying systems the only assumption we make on the state feedback control law is that it does not vanish as the states grow unboundedly for the observer we propose a design method based on the assumption that the system is transformable into a form affine in the unmeasured variable and a property of persistency of excitation this assumption covers but is not restricted to the common observability condition that the vector field multiplying the unmeasured variable is separated from zero for all state values our separation principle relies on stability results for cascades systems

the kth lower bases of primitive non powerful signed digraphs abstract in j shao l you h shan bound on the bases of irreducible generalized sign pattern matrices linear algebra appl 427 2007 285 300 the authors extended the concept of the base from powerful sign pattern matrices to non powerful irreducible sign pattern matrices recently the k th local bases and the k th upper bases which are generalizations of the bases of primitive non powerful signed digraphs were introduced in this paper we introduce a new parameter called the k th lower bases of primitive non powerful signed digraphs and obtain some bounds for it for some cases the bounds we obtain are best possible and the extremal signed digraphs are characterized respectively moreover we show that there exist gaps in the k th lower bases set of primitive non powerful signed digraphs

a survey on bases of sign pattern matrices abstract an n n sign pattern matrix has entries in 1 1 0 this paper surveys the following problems concerning the bases of sign pattern matrices base and base set of sign pattern matrices the generalizations of the base of sign pattern matrices other indices of sign pattern matrices several open problems are identified

quantization of coboundary lie bialgebras we show that any coboundary lie bialgebra can be quantized for this we prove that a etingof kazhdan quantization functors are compatible with lie bialgebra twists and b if such a quantization functor corresponds to an even associator then it is also compatible with the operation of taking coopposites we also use the relation between the etingof kazhdan construction of quantization functors and the alternative approach to this problem which was established in a previous work

bounded smooth state feedback and a global separation principle for non affine nonlinear systems abstract in this paper we show that a single input nonlinear system z 0 x g 1 x u g l x u l is globally asymptotically stabilizable by arbitrarily small smooth state feedback if unforced dynamics of are lyapunov stable and satisfies appropriate controllability like rank conditions characterized by the lie bracket of vector fields f0 x and g1 x an explicit smooth state feedback control law is constructed this in turn leads to an analytic resp smooth state feedback control law which solves the problem of global stabilization for analytic resp smooth systems x f x u based on these results we then establish a global separation principle for the system with an output finally we briefly discuss how the results can be extended to a multi input system

irreducible sign k potent sign pattern matrices abstract the sign pattern matrix a is called sign k potent if k is the smallest positive integer such that a k 1 a the structure of irreducible sign k potent pattern matrices was characterized by stuart et al j stuart c eschenbach s kirkland linear algebra appl 294 1999 85 92 we extend those results to the reducible case providing necessary conditions that characterize the structure of each off diagonal block of the frobenius normal form of a reducible sign k potent matrix

sontag s input to state stability condition and global stabilization using state detection this paper deals with the global stabilization problem for nonlinear system using state detection sufficient algebraic conditions for stabilization and detectability concerning particular classes of systems are included these conditions are based to a version of a lyapunov like condition proposed by sontag which guarantees input to state stability

on the period and base of a sign pattern matrix abstract a square sign pattern matrix a whose entries are or 0 is said to be powerful if all the powers a 1 a 2 a 3 are unambiguously defined for a powerful pattern a if a l a l p with l and p minimal then l is called the base of a and p is called the period of a we characterize irreducible powerful sign pattern matrices and investigate the period and base of a powerful sign pattern matrix we also consider some connections with real matrices and give some significant classes of powerful patterns

the minimum upper bound on the first ambiguous power of an irreducible nonpowerful ray or sign pattern abstract let a be an n n irreducible ray or sign pattern matrix if a is a sign pattern it is shown that either a is powerful or else a k has an ambiguous entry for some k n 2 2 n 2 and further sign patterns based on the wielandt graph show that this bound is the best possible if a is a ray pattern partial results for the same bound are given