change of rings and singularity categories abstract we investigate the behavior of singularity categories and stable categories of gorenstein projective modules along a morphism of rings the natural context to approach the problem is via change of rings that is the classical adjoint triple between the module categories in particular we identify conditions on the change of rings to induce functors between the two singularity categories or the two stable categories of gorenstein projective modules moreover we study this problem at the level of big singularity categories in the sense of krause 30 along the way we establish an explicit construction of a right adjoint functor between certain homotopy categories this is achieved by introducing the notion of 0 cocompact objects in triangulated categories and proving a dual version of bousfield s localization lemma we provide applications and examples illustrating our main results

normal convexity and equations over groups a subgroups of a grouph is said to be normal convex inh if for any subsetr s the natural maps r s h r his injective

stabilizing solutions of the h algebraic riccati equation the algebraic riccati equation studied in this paper is related to the suboptimal state feedback h control problem it is parametrized by the h norm bound we want to achieve the objective of this paper is to study the behavior of the solution to the riccati equation as a function of it turns out that a stabilizing solution exists for all but finitely many values of larger than some a priori determined bound on the other hand for values smaller than there does not exist a stabilizing solution the finite number of exception points can be characterized as switching points where eigenvalues of the stabilizing symmetric solution can switch from negative to positive with increasing after the final switching point the solution will be positive semidefinite we obtain the following interpretation the riccati equation has a stabilizing solution with k negative eigenvalues if and only if there exists a static feedback such that the closed loop transfer matrix has k unstable poles and an l norm strictly less than

the groups gm n p abstract we determine except for two outstanding cases for which values of m n and p the group g m n p defined by the presentation a b c a 2 b 2 a b 2 c 2 a c m b c n a b c p is finite these groups formed one of three families of groups first studied systematically by coxeter in 1939

brief paper the extended periodic lyapunov lemma in this paper the so called lyapunov lemma for linear systems with periodically time varying coefficients is dealt with by this lemma the asymptotic stability of a given periodic system is linked with the existence of a periodic positive definite solution of the associated lyapunov equation under the assumption on the system being observable by relaxing the observability assumption to detectability an extended version of the lemma is obtained where the system stability is linked with the existence of a periodic positive semidefinite solution of the lyapunov equation

the grothendieck duality theorem via bousfield s techniques and brown representability grothendieck proved that if f x y is a proper morphism of nice schemes then rf has a right adjoint which is given as tensor product with the relative canonical bundle the original proof was by patching local data deligne proved the existence of the adjoint by a global argument and verdier showed that this global adjoint may be computed locally in this article we show that the existence of the adjoint is an immediate consequence of brown s representability theorem lit follows almost as imme diately by smashing arguments that the adjoint is given by tensor product with a dualising complex verdier s base change theorem is an easy conse quence department of mathematics university of virginia charlottesville virginia 22903 e mail address an3rqvirginia edu this content downloaded from 157 55 39 27 on wed 07 sep 2016 06 26 19 utc all use subject to http about jstor org terms

stable homotopy as a triangulated functor let j i be the homotopy category of all finite spectra the homotopy category of all spectra and y 89 be the homotopy category of all prime to 2 spectra let h y abelian groups be a homological functor i e a generalized homology theory a natural question is whether h can be lifted to a triangulated functor is there a functor t y d abelian groups so that composed with the usual homology functor on d abelian groups is h here d abelian groups d z is the derived category of z modules if h is ordinary homology the answer is clearly yes after all the ordinary homology of a space is given by the singular chain complex almost by definition one has a lifting by the brown representability theorem the general statement follows from the special case h h the stable homotopy functor let us be a little more precise

graded modules of gelfand kirillov dimension one over three dimensional artin schelter regular algebras abstract let a be a three dimensional artin schelter regular algebra we give a description of the category of finitely generated a modules of gelfand kirillov dimension one modulo those of finite dimension over the ground field the proof is based upon a result by gabriel which says that locally finite categories can be described by module categories over topological rings

intersection theory on non commutative surfaces consider a non commutative algebraic surface x and an effective divisor y on x as defined by van den bergh we show that the riemann roch theorem the genus formula and the self intersection formula from classical algebraic geometry generalize to this setting we also apply our theory to some special cases including the blow up of x in a point and show that the self intersection of the exceptional divisor is 1 this is used to give an example of a non commutative surface with a commutative p1 which cannot be blown down because its self intersection is 1 rather than 1 we also get some results on hilbert polynomials of modules on x

bounded rank subgroups of coxeter groups artin groups and one relator groups with torsion we obtain a number of results regarding the freeness of subgroups of coxeter groups artin groups and one relator groups with torsion in the case of coxeter groups we also obtain results on quasiconvexity and subgroup separability

non orientable surface plus one relation groups recently dicks linnell determined the l 2 betti numbers of the orientable surfaceplus one relation groups and their arguments involved some results that were obtained topologically by hempel and howie using algebraic arguments we now extend all these results of hempel and howie to a larger class of two relator groups and we then apply the extended results to determine the l 2 betti numbers of the non orientable surface plus one relation groups 2000 mathematics subject classification primary 20f05 secondary 16s34 20j05

analysis of the periodic lyapunov and riccati equations via canonical decomposition a new proof is given of the following result a periodic riccati equation admits a unique positive semidefinite periodic solution and the solution is stabilizing if and only if the underlying system is stabilizable and detectable the proof hinges on the decomposition of the riccati equation induced by the system canonical decomposition the solution structure is therefore naturally pointed out

the geometry of arithmetic noncommutative projective lines abstract let k be a perfect field and let k k be a finite extension of fields an arithmetic noncommutative projective line is a noncommutative space of the form proj s k v where v be a k central two sided vector space over k of rank two and s k v is the noncommutative symmetric algebra generated by v over k defined by m van den bergh 26 we study the geometry of these spaces more precisely we prove they are integral we classify vector bundles over them we classify them up to isomorphism and we classify isomorphisms between them using the classification of isomorphisms we compute the automorphism group of an arithmetic noncommutative projective line

serre finiteness and serre vanishing for non commutative p1 bundles abstract suppose x is a smooth projective scheme of finite type over a field k e is a locally free o x bimodule of rank 2 a is the non commutative symmetric algebra generated by e and proj a is the corresponding non commutative p 1 bundle we use the properties of the internal hom functor h om gr a to prove versions of serre finiteness and serre vanishing for proj a as a corollary to serre finiteness we prove that proj a is ext finite this fact is used in i mori j pure appl algebra in press to prove that if x is a smooth curve over speck proj a has a riemann roch theorem and an adjunction formula

a translation principle for the four dimensional sklyanin algebras abstract in this paper we prove a translation principle for the central quotients of four dimensional sklyanin algebras which is analogous to the translation principle for semi simple lie algebras in the course of the proof we construct an elliptic analog of the sheaf of differential operators on2 1 the translation principle may be used to construct the fat points of a four dimensional non pi sklyanin algebra

structure of words with short 2 length in a free product of groups abstract howie and duncan observed that a word in a free product with length at least two which is not a proper power and involves no letter of order two can be decomposed as a product of two cyclic subwords each of which is uniquely positioned using this property they proved various important results about a one relator product of groups with such word as the relator in this paper we show that similar results hold in a more general setting where we allow a certain number of elements of order two

an asphericity conjecture and kaplansky problem on zero divisors suppose a group representationh a r is aspherical x a w a x is a word in alphabet a x 1with nonzero sum of exponents onx and the grouphnaturally embeds ing a x r w a x it is conjectured that the presentationg a x r w a x is aspherical if and only ifgis torsion free it is proven that if this conecture is false andg a x r w a x is a counterexample then the integral group ring z g of torsion free groupgwill contain zero divisors some special cases when this conjecture holds are also indicated

implicit function theorem over free groups abstract we introduce the notion of a regular quadratic equation and a regular ntq system over a free group we prove the results that can be described as implicit function theorems for algebraic varieties corresponding to regular quadratic and ntq systems we will also show that the implicit function theorem is true only for these varieties in algebraic geometry such results would be described as lifting solutions of equations into generic points from the model theoretic view point we claim the existence of simple skolem functions for particular formulas over free groups proving these theorems we describe in details a new version of the makanin razborov process for solving equations in free groups we also prove a weak version of the implicit function theorem for ntq systems which is one of the key results in the solution of the tarski s problems about the elementary theory of a free group we call it the parametrization theorem

genericity the arzhantseva ol shanskii method and the isomorphism problem for one relator groups we show that the isomorphism problem is solvable in at most exponential time for a class of one relator groups which is exponentially generic in the sense of ol shanskii this is obtained by applying the arzhantseva ol shanskii graph minimization method to prove the general result that for fixed integers m 2 and n 1 there is an exponentially generic class of non free m generator n relator groups with the property that there is only one nielsen equivalence class of m tuples which generate a non free subgroup in particular every m generated subgroup in such a generic group g is either free or is equal to g itself and such groups are thus co hopfian these results are obtained by elementary methods without using the deep results of sela about co hopficity and the isomorphism problem for torsion free hyperbolic groups

finite generalized triangle groups we give an almost complete classification of those generalized tri angle groups that are finite building on previous results of baumslag morgan and shalen 1 conder 4 rosenberger 12 and levin and rosenberger 11 there are precisely two groups for which we cannot decide whether or not they are finite

non commutative mori contractions and p1 bundles abstract we give a method for constructing maps from a non commutative scheme to a commutative projective curve with the aid of artin zhang s abstract hilbert schemes this is used to construct analogues of the extremal contraction of a k negative curve with self intersection zero on a smooth projective surface this result will hopefully be useful in studying artin s conjecture on the birational classification of non commutative surfaces as a non trivial example of the theory developed we look at non commutative ruled surfaces and more generally at non commutative p 1 bundles we show in particular that non commutative p 1 bundles are smooth have well behaved hilbert schemes and we compute its serre functor we then show that non commutative ruled surfaces give examples of the aforementioned non commutative mori contractions

l 2 betti numbers of one relator groups we determine the l 2 betti numbers of all one relator groups and all surface plus one relation groups we also obtain some information about the l 2 cohomology of left orderable groups and deduce the non l 2 result that in any left orderable group of homological dimension one all two generator subgroups are free

irreducible affine varieties over a free group i irreducibility of quadratic equations and nullstellensatz the object of this paper which consists of two parts is to describe irreducible varieties over free groups and to characterize finitely generated fully residually free groups we prove that any variety over a free group f can be defined by a finite number of systems of equations s 1 in triangular form where quadratic words play the role of leading terms algebraically irreducible varieties are exactly the varieties whose coordinate groups are fully residually f the crucial point of the classification of fully residually free groups is to prove that the coordinate groups of irreducible varieties are embeddable into lyndon s free z x group f since every finitely generated fully residually free group is a free factor of the coordinate group of an irreducible variety and the group f is fully residually f we obtain a characterization of finitely generated fully residually free groups as subgroups of f the group f and its subgroups have been studied extensively during the last several years in particular every finitely generated subgroup of f hence every finitely generated fully residually free group can be obtained from free abelian groups of finite rank by finitely many free products with amalgamation and hnn extensions of the type where amalgamated and associated subgroups are free abelian of finite rank in particular this implies that every finitely generated fully residually free group is finitely presented there are three parts to this paper algebraic geometry over free groups the theory of free exponential groups and makanin razborov s machinery to deal with equations over free groups 11 14 13 the algebraic geometry approach has been shown to be very usefull in dealing with equations over groups it provides necessary topological means and a method to transcribe geometric notions into pure group theoretic language following baumslag myasnikov remeslennikov 1 we use the standard algebraic geometry notions such as variety zariski topology irreducibility of varieties radicals and coordinate groups some of the ideas of the algebraic geometry approach go back to r lyndon 9 e rips j stallings 16

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

on balanced presentations of the trivial group to solve problems of magnus and stallings stated in the 1960s we construct a balanced presentation of the trivial group such that no defining relator of this presentation could be replaced by a free generator and the group given by the altered presentation would still be trivial some other related problems on balanced presentations of the trivial group are discussed

finite escapes and convergence properties of guaranteed cost robust filters the design of robust filters for linear time varying systems subject to parameter perturbations is considered the design is based on the minimization of an upper bound for the error covariance the computation of the filter matrices calls for the solution of a pair of differential riccati equations parameterized by a scalar design parameter the main results of the paper concern the asymptotic properties of the filter when it is applied to time invariant systems differently from the standard kalman filter the robust filter may abruptedly cease to exist owing to the presence of finite escape times in the associated riccati equations nevertheless provided that the parameter perturbations do not affect the unstable dynamics convergence to a stable steady state filter can always be ensured by properly adjusting the scalar design parameter since such an adjustment might deteriorate the steady state performance a time varying strategy for the tuning of the scalar parameter is developed in order to guarantee both convergence and performance

serre duality for non commutative mathbb p 1 bundles let x be a smooth scheme of finite type over a field k let e be a locally free o x bimodule of rank n and let a be the non commutative symmetric algebra generated by e we construct an internal hom functor h omgra on the category of graded right a modules when e has rank 2 we prove that a is gorenstein by computing the right derived functors of hom gr a o x when x is a smooth projective variety we prove a version of serre duality for proja using the right derived functors of lim hom gra a a n

noncommutative tsen s theorem in dimension one abstract let k be a field in this paper we find necessary and sufficient conditions for a noncommutative curve of genus zero over k to be a noncommutative p 1 bundle this result can be considered a noncommutative one dimensional version of tsen s theorem by specializing this theorem we show that every arithmetic noncommutative projective line is a noncommutative curve and conversely we characterize exactly those noncommutative curves of genus zero which are arithmetic we then use this characterization together with results from 9 to address some problems posed in 4

the group g t e when g is torsion free abstract in this paper we discuss classes of nontrivial elements in the group with a relative presentation g t e where g is torsion free and e is solvable over g

cyclomonotonicity and stabilizability properties of solutions of the difference periodic riccati equation the kalman filter associated with a discrete time linear t periodic system is tested the problem considered is that of selecting an initial covariance matrix such that the periodic filter based on the first t values of the kalman filter gain is stabilizing sufficient conditions are given that hinge on the cyclomonotonicity of the solution of the periodic riccati equation potential applications are found in filter design quasi linearization techniques for the periodic riccati equation and the design of receding horizon control strategies for periodic and multirate systems when specialized to time invariant systems the results give rise to new sufficient conditions for the cyclomonotonicity of the solutions of the time invariant riccati equation and the existence of periodic stabilizing feedback

inertia theorems for the periodic liapunov equation and periodic riccati equation the inertia theorems for the time invariant liapunov equation and time invariant riccati equation are generalized to the case were the coefficient matrices are periodically time varying

guaranteed cost certification for discrete time linear switched systems with a dwell time this technical note studies the guaranteed cost of a quadratic criterion associated with a linear discrete time switched system for all the set of admissible switching laws the admissible switching laws are here the ones exhibiting a dwell time the approach provided here is to design an upper bound and a parametrized family of lower bounds of the guaranteed cost as close as possible in order to obtain a certification of the guaranteed cost the upper bound is determined via a switched lyapunov function and the lower bounds are obtained via the numerical computation of the cost induced by particular periodic switching laws the features of the proposed approach are illustrated by a numerical example

inertia theorems for the periodic lyapunov difference equation and periodic riccati difference equation abstract the periodic lyapunov difference equation plde and periodic riccati difference equation prde are dealt with the inertia i e the number of positive null and negative eigenvalues of any symmetric periodic solution of such equations is linked with the pattern of eigenvalues of the monodromy matrix associated with the open loop for plde or closed loop for prde underlying systems different results are obtained by imposing requirements with decreasing strength to the original system precisely assumptions of observability reconstructibility and detectability are successively introduced some results are also given for the particular case of positive semidefinite periodic solutions

branched coverings of 2 complexes and diagrammatic reducibility the condition that all spherical diagrams in a 2 complex be reducible is shown to be equivalent to the condition that all finite branched covers be aspherical this result is related to the study of equations over groups furthermore large classes of 2 complexes are shown to be diagrammatically reducible in the above sense in particular every 2 complex has a subdivision which admits a finite branched cover which is diagrammatically reducible a consequence of the classical riemann hurwitz formula for riemann surfaces 6 p 301 is the fact that a finite branched cover of an aspherical riemann surface is also aspherical however the property of asphericity of a 2 complex implying asphericity of finite branched covers fails for general 2 complexes in fact we shall prove theorem 4 5 below that this property holds precisely for those 2 complexes which are diagrammatically reducible a 2 complex x is said to be diagrammatically reducible abbreviated dr x if for every combinatorial map f s2 x of the 2 sphere into x a spherical diagram in x there is a pair of faces in the domain with an edge in common which are mapped mirrorwise by f across that edge a closely related notion was introduced by lyndon and schupp 13 but this notion appears in its present form in a paper by sieradski 14 in that same paper sieradski asked a question which would have implied that all classical knot complements in the 3 sphere are homotopy equivalent to diagrammatically reducible 2 complexes we shall prove this latter assertion theorem 6 5 making use of our amalgamation theorem for diagrammatic reducibility theorem 5 4 the characterization of diagrammatic reducibility in terms of branched covers llows from a lifting theorem which may be of general interest if f x y is a combinatorial map of combinatorial 2 complexes then f induces a map of the link complexes of vertices lf lx ly the star graphs or coinitial graph in group theoretical language the map f is said to be reduced if lf is an immersion we prove theorem 4 2 th t if f x y is a reduced map of 2 complexes with x finite then there is a commutative diagram see below where qt z y is a finite branched cover and where g is injective off the zero skeleton of x this result is received by the editors july 9 1986 1980 muthenluticssulviectclussificwtios 1985 retvisios primary20f05 57m12 secondary57m20 1987 american mathematical scoclety 0002 9947 87 1 0 25 pel page

stable commutator length in baumslag solitar groups and quasimorphisms for tree actions this paper has two parts on baumslag solitar groups and on general g trees in the first part we establish bounds for stable commutator length scl in baumslag solitar groups for a certain class of elements we further show that scl is computable and takes rational values we also determine exactly which of these elements admit extremal surfaces in the second part we establish a universal lower bound of 1 12 for scl of suitable elements of any group acting on a tree this is achieved by constructing efficient quasimorphisms calculations in the group bs 2 3 show that this is the best possible universal bound thus answering a question of calegari and fujiwara we also establish scl bounds for acylindrical tree actions returning to baumslag solitar groups we show that their scl spectra have a uniform gap no element has scl in the interval 0 1 12

species and non commutative p1 s over non algebraic bimodules abstract we study non commutative projective lines over not necessarily algebraic bimodules in particular we give a complete description of their categories of coherent sheaves and show they are derived equivalent to certain bimodule species this allows us to classify modules over these species and thus generalize and give a geometric interpretation for results of c ringel 21

a note on the maximal solution of the periodic riccati equation the periodic symmetric solutions of the periodic riccati differential equation associated with the filtering problem are considered by the authors it is proven that under the sole assumption of detectability there exists a maximal solution moreover such a solution turns out to be strong i e the characteristic multipliers of the associated closed loop system belong to the closed unit disk the proof relies on an iterative linearization technique which calls for a sequence of periodic lyapunov equations similar results are given for the minimal solution

ordering coproducts of groups and semigroups abstract a a vinogradov proved that a coproduct of orderable groups is orderable and r e johnson extended this result to semigroups a simpler version of their proof is given and results are obtained on orderability of group coproducts with amalgamation in particular it is shown that the coproduct of two or more copies of an orderable group h with amalgamation of a subgroup t h is orderable if and only if t is the difference kernel of a pair of homomorphisms from h into another orderable group equivalently if and only if t is convex under some one sided group ordering of h for coproducts of families of nonisomorphic overgroups of a group t partial results are obtained some related results are proved for one sided orderability

scl in graphs of groups let g be a group acting on a tree with cyclic edge and vertex stabilizers then stable commutator length scl is rational in g furthermore scl varies predictably and converges to rational limits in so called surgery families this is a homological analog of the phenomenon of geometric convergence in hyperbolic dehn surgery

maximal solution of a certain class of periodic riccati differential equations this paper deals with a generalized version of the well known periodic riccati differential equation which arises in certain optimal filtering problems for linear periodic systems a monotonicity property for symmetric periodic equilibrium solutions is established and it is proved that there exists a maximal symmetric periodic equilibrium solution

a new test for asphericity and diagrammatic reducibility of group presentations we present a new test for studying asphericity and diagrammatic reducibility of group presentations our test can be applied to prove diagrammatic reducibility in cases where the classical weight test fails we use this criterion to generalize results of j howie and s m gersten on asphericity of lots and of adian presentations and derive new results on solvability of equations over groups we also use our methods to investigate a conjecture of s v ivanov related to kaplansky s problem on zero divisors we strengthen ivanov s result for locally indicable groups and prove a weak version of the conjecture

the difference periodic ricati equation for the periodic prediction problem gives a comprehensive treatment of several important aspects of the discrete time periodic riccati equation dpre arising from the prediction problem for linear discrete time periodic systems the authors analyze the symmetric periodic positive semidefinite spps solution of the dpre under appropriate assumptions of stabilizability and detectability of the periodic system among the results obtained are necessary and sufficient conditions for the existence and uniqueness of the spps solution and the stability of the resulting closed loop system some of these results can be seen as extensions of the corresponding results for the time invariant case however a number of them contain contributions to the time invariant case as well the paper also gives a numerical algorithm based on an iterative linearization procedure for computing the spps solution the algorithm is a periodic version of kleinman s algorithm for the time invariant case

stabilizability and detectability of linear periodic systems abstract in this paper four characterizations of stabilizability and detectability of linear periodic systems are considered two of them look as natural extensions of the classical definitions given for time invariant systems the remaining two are modal characterizations which turn out to be useful in the analysis of the periodic lyapunov and riccati equations it is shown that all these notions of stabilizability and detectability are in fact equivalent to each other one of the various definitions calls for the existence of the kalman canonical decomposition of periodic systems this issue is addressed in the appendix

algebraic geometry over groups i algebraic sets and ideal theory abstract the object of this paper which is the first in a series of three is to lay the foundations of the theory of ideals and algebraic sets over groups

non noetherian groups and primitivity of their group algebras abstract we prove that the group algebra kg of a group g over a field k is primitive provided that g has a non abelian free subgroup with the same cardinality as g and that g satisfies the following condition for each subset m of g consisting of a finite number of elements not equal to 1 and for any positive integer m there exist distinct a b and c in g so that if x 1 1 g 1 x 1 x m 1 g m x m 1 where g i is in m and x i is equal to a b or c for all i between 1 and m then x i x i 1 for some i this generalizes results of 1 9 17 and 18 and proves that for every countably infinite group g satisfying kg is primitive for any field k we use this result to determine the primitivity of group algebras of one relator groups with torsion

subspaces of non commutative spaces this paper concerns the closed points closed subspaces open subspaces weakly closed and weakly open subspaces and effective divisors on a non commutative space

the second bounded cohomology of an amalgamated free product of groups we study the second bounded cohomology of an amalgamated free product of groups and an hnn extension of a group as an application we show that a group with infinitely many ends has infinite dimensional second bounded cohomology

periodic solutions of periodic riccati equations for periodically time varying matrix riccati equations controllability and observability in the usual sense are shown to be sufficient for the existence of a unique positive definite periodic solution

on locally indicable groups a group is said to be locally indicable if each of its non trivial finitely generated subgroups has the infinite cyclic group as a homomorphic image such groups were studied by higman 9 in connection with the zero divisor and unit problems for group rings more recently they have arisen i 2 12 in the study of equations over groups in 12 i proved a freiheitsatz for locally indicable groups this has been proved independently by brodskii 2 and short 22 the present paper arises from an investigation of a question put to me by s j pride whether torsionfree 1 relator groups are locally indicable the question was raised originally by baumslag i 1 problem 19 and an affirmative solution has recently been announced by brodski 2 as a consequence the group algebra rg of a torsion free 1 relator group g over an integral domain r has no non trivial zero divisors and no non trivial units using higman s results 9 the first fact was also proved by lewin and lewin 16 by embedding rg in a division ring the second appears to be new as was pointed out to me by k a brown a second consequence is that no 1 relator group has a non trivial finitely generated perfect subgroup which answers 1 problem 7 and 10 question 1 in fact using the freiheitsatz and the tower method described in 12 it is possible to prove the following general version of brodskii s theorem

stability and inertia abstract the purpose of this paper is to present a brief overview of matrix stability and inertia theory a few applications of inertia and stability theorems and a nonspectral implicit matrix equation method for determining stability and inertia of a nonhermitian matrix are also presented inter relationships between different theorems are explicitly stated whenever appropriate the paper concludes with some problems for future research in this area

on systems of equations over free products of groups using an analogue of the makanin razborov diagrams we give a description of the solution set of systems of equations over an equationally noetherian free product of groups g equivalently we give a parametrisation of the set hom h g of all homomorphisms from a finitely generated group h to g furthermore we show that every algebraic set over g can be decomposed as a union of finitely many images of algebraic sets of ntq systems if the universal horn theory of g the theory of quasi identities is decidable then our constructions are effective 2010 elsevier inc

the injective spectrum of a noncommutative space abstract for a noncommutative space x we study inj x the set of isomorphism classes of indecomposable injective x modules in particular we look at how this set suitably topologized can be viewed as an underlying spectrum for x as applications we discuss noncommutative notions of irreducibility and integrality and a way of associating an integral subspace of x to each element of inj x which behaves like a weak point

on the asphericity of ribbon disc complements the complement of a ribbon n disc in the 71 2 ball has a 2 dimensional spine which shares some of the combinatorial properties of classical knot complement spines it is an open question whether such 2 complexes are always aspherical to any ribbon disc we associate a labelled oriented tree from which the homotopy type of the complement can be recovered and we prove asphericity in certain special cases described by conditions on this tree our main result is that the complement is aspherical whenever the associated tree has diameter at most 3

h controllability and observability of linear periodic systems with reference to linear periodic systems the classical notion of controllability introduced by kalman k controllability is shown to be equivalent to the characterization of controllability h controllability first proposed by hewer as a natural extension to the periodic case of an eigenvalue eigenvector controllability condition independently introduced by several authors for time invariant systems only the proof of such an equivalence also leads to the conclusion that if a t periodic system s is k controllable and k is the degree of the minimal polynomial of the monodromy matrix associated with s then at any time t s is controllable over t t kt since k is lower than or equal to the system order a well known result due to brunovsky is slightly strengthened by duality the corresponding observability results follow

lyapunov and riccati equations periodic inertia theorems the lyapunov and riccati differential equations with periodically time varying coefficients are considered under the assumption of detectability of the underlying periodic system two inertia theorems are provided linking the inertia of the solution to the one of the so called monodromy matrix

stable commutator length is rational in free groups for any group there is a natural pseudo norm on the vector space b h 1 of real homogenized group 1 boundaries called the stable commutator length norm this norm is closely related to and can be thought of as a relative version of the gromov pseudo norm on ordinary homology we show that for a free group the unit ball of this pseudo norm is a rational polyhedron it follows that the stable commutator length in free groups takes on only rational values moreover every element of the commutator subgroup of a free group rationally bounds an injective map of a surface group the proof of these facts yields an algorithm to compute the stable commutator length in free groups using this algorithm we answer a well known question of bavard in the negative constructing explicit examples of elements in free groups whose stable commutator length is not a half integer

estimates of asymptotic trajectory bounds in digital implementations of linear feedback control systems estimates are established for trajectory bounds which are easy to calculate applicable to general controller topologies and dependent on parameters i e eigenvalues and eigenvectors of the idealized linear system i e the control system model neglecting quantizers the approach utilizes the comparison principle in the lyapunov theory and the properties of minimal matrix norms the applicability of the results are demonstrated by means of a specific example

inertia theorems for operator lyapunov inequalities we study operator lyapunov inequalities and equations for which the in1nitesimal generator is not necessarily stable but it satis1es the spectrum decomposition assumption and it has at most 1nitely many unstable eigenvalues moreover the input or output operators are not necessarily bounded but are admissible we prove an inertia result under mild conditions we show that the number ofunstable eigenvalues ofthe generator is less than or equal to the number ofnegative eigenvalues ofthe self adjoint solution ofthe operator lyapunov inequality c 2001 elsevier science b v all rights reserved

a remarkable family of left ordered groups central extensions of hecke groups abstract we provide an infinite family of left ordered groups that have a positive cone that is generated by two elements as a semigroup this family corresponds to that of certain central extensions of hecke groups and includes the klein bottle group and the braid group b 3 using the classical convex extension flipping procedure on each of the groups of this family we define an ordering sharing many properties with dehornoy s several related questions and problems are addressed

quadratic equations in hyperbolic groups are np complete we prove that in a torsion free hyperbolic group the length of the value of each variable in a minimal solution of a quadratic equation q 1 is bounded by njqj 4 for an orientable equation and by njqj 16 for a non orientable equation where jqj is the length of the equation and the constant n can be computed we show that the problem whether a quadratic equation in has a solution is np complete and that there is a pspace algorithm for solving arbitrary equations in we also give a slightly larger bound for minimal solutions of quadratic equations in a toral relatively hyperbolic group