quadratic stabilizability of uncertain linear systems existence of a nonlinear stabilizing control does not imply existence of a linear stabilizing control this note is concerned with the problem of stabilizing an uncertain linear system via state feedback control an uncertain system which admits a stabilizing state feedback control and some associated quadratic lyapunov function is said to be quadratically stabilizable in a number of recent papers conditions are given under which quadratic stabilizability via nonlinear control implies quadratic stabilizability via linear control these papers restrict the manner in which the uncertain parameters are permitted to enter structurally into the state equation in order to establish this result this note presents an example which shows that this implication is not true for more general uncertain linear systems to this end we describe an uncertain linear system which is quadratically stabilizable via nonlinear control but not quadratically stabilizable via linear control

k theory of log schemes ii log syntomic k theory abstract we prove that suitably truncated topological log syntomic etale homotopy k theory of proper semistable schemes in mixed characteristic surjects onto the kummer log etale p adic k theory of the generic fiber as a corollary we get that the log syntomic etale cohomology is a direct factor of the log syntomic etale cohomology of homotopy k theory sheaves the proofs use p adic hodge theory computations of p adic nearby cycles

algebraic cycles and higher k theory here g x is the grothendieck group of coherent sheaves on x 13 gr refers to the graded group defined by the y filtration on g x cf kratzer 14 soult 20 and ch x is the chow group of codimension i algebraic cycles defined by fulton 9 the left hand isomorphism is a formal consequence of the existence of a i structure on g x while the existence of r is the central theme of the b f m rr theorem the main purpose of this paper is to define a theory of higher chow groups ch x n n 2 0 so as to obtain isomorphisms

artin tate motivic sheaves with finite coefficients over an algebraic variety we propose a construction of a tensor exact category fxm of artin tate motivic sheaves with finite coefficients z m over an algebraic variety x over a field k of characteristic prime to m in terms of etale sheaves of z m modules over x among the objects of fxm in addition to the tate motives z m j there are the cohomological relative motives with compact support mccm y x of varieties y quasi finite over x exact functors of inverse image with respect to morphisms of algebraic varieties and direct image with compact supports with respect to quasi finite morphisms of varieties y x act on the exact categories fxm assuming the existence of triangulated categories of motivic sheaves dm x z m over algebraic varieties x over k and a weak version of the six operations in these categories we identify fxm with the exact subcategory in dm x z m consisting of all the iterated extensions of the tate twists mccm y x j of the motives mccm y x an isomorphism of the z m modules ext between the tate motives z m j in the exact category fxm with the motivic cohomology modules predicted by the beilinson lichtenbaum etale descent conjecture recently proved by voevodsky rost et al holds for smooth varieties x over k if and only if the similar isomorphism holds for artin tate motives over fields containing k when k contains a primitive m root of unity the latter condition is equivalent to a certain koszulity hypothesis as shown in our previous paper positselski mixed artin tate motives with finite coefficients mosc math j 11 2011 317 402

the group sk 2 of a biquaternion algebra m rost has proved the existence of an exact sequence relating the group sk1 kernel of the reduced norm for k1 of a biquaternion algebra d whose center is a field f with galois cohomology groups of f in this paper we relate the group sk2 kernel of the reduced norm for k2 of d with galois cohomology of f through an exact sequence

rigidity for pseudo pretheories assume f is a homotopy invariant pseudo pretheory with torsion values and x is a smooth scheme of finite type over a field k we show for certain field extensions k k the map f x f xk is an isomorphism

hermitian k theory of the integers rognes and weibel used voevodsky s work on the milnor conjecture to deduce the strong dwyer friedlander form of the lichtenbaum quillen conjecture at the prime 2 in consequence the 2 completion of the classifying space for algebraic k theory of the integers z 1 2 can be expressed as a fiber product of well understood spaces bo and bgl f3 over bu similar results are now obtained for hermitian k theory and the classifying spaces of the integral symplectic and orthogonal groups for the integers z 1 2 this leads to computations of the 2 primary hermitian k groups and affirmation of the lichtenbaum quillen conjecture in the framework of hermitian k theory

on a convex parameter space method for linear control design of uncertain systems this paper presents a new procedure for continuous and discrete time linear control systems design it consists of the definition of a convex programming problem in the parameter space that when solved provides the feedback gain one of the most important features of the procedure is that additional design constraints are easily incorporated in the original formulation yielding solutions to problems that have raised a great deal of interest within the last few years this is precisely the case of the decentralized control problem and the quadratic stabilizability problem of uncertain systems with both dynamic and input uncertain matrices in this last case necessary and sufficient conditions for the existence of a linear stabilizing gain are provided and to the authors knowledge this is one of the first numerical procedures able to handle and solve this interesting design problem for high order continuous time or discrete time linear models the theory is illustrated by examples

the quadratic matrix inequality in singular h control with state feedback in this paper the standard h infty control problem using state feedback is considered given a linear time invariant finite dimensional system this problem consists of finding a static state feedback such that the resulting closed loop transfer matrix has h infty norm smaller than some a priori given upper bound in addition it is required that the closed loop system is internally stable conditions for the existence of a suitable state feedback are formulated in terms of a quadratic matrix inequality reminiscent of the dissipation inequality of singular linear quadratic optimal control where the direct feedthrough matrix of the control input is injective the results presented here specialize to known results in terms of solvability of a certain indefinite algebraic riccati equation

the homotopy fixed point theorem and the quillen lichtenbaum conjecture in hermitian k theory abstract we settle two conjectures for computing higher grothendieck witt groups also known as hermitian k groups of noetherian schemes x under some mild conditions it is shown that the comparison map from the hermitian k theory of x to the homotopy fixed points of k theory under the natural z 2 action is a 2 adic equivalence we also prove that the mod 2 comparison map between the hermitian k theory of x and its etale version is an isomorphism on homotopy groups in the same range as for the quillen lichtenbaum conjecture in k theory applications compute higher grothendieck witt groups of complex algebraic varieties and rings of 2 integers in number fields and hence values of dedekind zeta functions

algebraic geometry of topological spaces i we use techniques from both real and complex algebraic geometry to study k theoretic and related invariants of the algebra c x of continuous complex valued functions on a compact hausdorff topological space x for example we prove a parameterized version of a theorem by joseph gubeladze we show that if m is a countable abelian cancellative torsion free semi normal monoid and x is contractible then every finitely generated projective module over c x m is free the particular case m mathbb n 0 n gives a parameterized version of the celebrated theorem proved independently by daniel quillen and andrei suslin that finitely generated projective modules over a polynomial ring over a field are free the conjecture of jonathan rosenberg which predicts the homotopy invariance of the negative algebraic k theory of c x follows from the particular case m mathbb z n we also give algebraic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on c algebras and for a homology theory of commutative algebras to vanish on c algebras these criteria have numerous applications for example the vanishing criterion applied to nil k theory implies that commutative c algebras are k regular as another application we show that the familiar formulas of hochschild kostant rosenberg and loday quillen for the algebraic hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic hochschild and cyclic homology of c x applications to the conjectures of be linson soule and farrell jones are also given

stabilizability of linear time varying and uncertain linear systems feedback stabilization of linear time varying and uncertain linear systems is considered it is proved that given a stabilizing dynamic linear state feedback controller one can always construct a stabilizing nondynamic linear state feedback controller a similar result is shown for uncertain linear systems a linear time varying system can be stabilized by dynamic output feedback if and only if it admits a coprime factorization

the methods of vorobyev and lanczos abstract this paper deals with the method of moments of vorobyev which is a particular case of galerkin s method this method is generalized in different ways and some of its properties are studied in detail as an application it is showed that the lanczos method for solving systems of linear equations can be derived from it and that some generalizations could be obtained

topological hypercovers and 1 realizations we show that if u is a hypercover of a topological space x then the natural map hocolim u x is a weak equivalence this fact is used to construct topological realization functors for the 1 homotopy theory of schemes over real and complex fields in an appendix we also prove a theorem about computing homotopy colimits of spaces that are not cofibrant

rigidity for equivariant pseudo pretheories abstract we prove versions of the suslin and gabber rigidity theorems in the setting of equivariant pseudo pretheories of smooth schemes over a field with an action of a finite group examples include equivariant algebraic k theory presheaves with equivariant transfers equivariant suslin homology and bredon motivic cohomology

necessary and sufficient conditions for quadratic controllability of a class of uncertain systems we introduce a controllability notion for uncertain systems namely quadratic controllability for a specific class of norm bounded uncertain systems we show that this notion is equivalent to controllability of a nominal system and the satisfaction of a certain structural condition by the uncertainty for systems in which the uncertain parameters affect all control inputs this structural condition is simply the familiar matching condition commonly encountered in the literature on robust and adaptive control of uncertain systems

homology of gln injectivity conjecture for gl4 the homology of gln r and sln r is studied where r is a commutative ring with many units our main theorem states that the natural map h4 gl3 r k h4 gl4 r k is injective where k is a field with char k 2 3 for an algebraically closed field f we prove a better result namely h 4 rm gl 3 f mathbb z to h 4 rm gl 4 f mathbb z is injective we will prove a similar result replacing gl by sl this is used to investigate the indecomposable part of the k group k4 r

lanczos versus subspace iteration for solution of eigenvalue problems in this paper we consider solution of the eigen problem in structural analysis using a recent version of the lanczos method and the subspace method the two methods are applied to examples and we conclude that the lanczos method has advantages which are too good to overlook

etale realization on the a1 homotopy theory of schemes abstract we compare friedlander s definition of the etale topological type for simplicial schemes to another definition involving realizations of pro simplicial sets this can be expressed as a notion of hypercover descent for etale homotopy we use this result to construct a homotopy invariant functor from the category of simplicial presheaves on the etale site of schemes over s to the category of pro spaces after completing away from the characteristics of the residue fields of s we get a functor from the morel voevodsky a 1 homotopy category of schemes to the homotopy category of pro spaces

quadratic controllability strong controllability and a related output feedback property quadratic controllability and strong controllability are the two system properties to be discussed for a special class of norm bounded uncertain linear systems the main contribution of this paper is twofold first we show that it is possible to reduce the problem of checking quadratic controllability or strong controllability of a given uncertain system to the same problem for a reduced order subsystem which we call the essential subsystem then we use this result to examine a related output feedback property specifically for a given uncertain system we will answer the following question what condition is necessary and sufficient for the existence of a dynamic output feedback controller such that the states of the closed loop system converge to the origin with an arbitrarily fast rate our aim is to provide an insight and better understanding of uncertain systems through these results

rigidity for linear framed presheaves and generalized motivic cohomology theories abstract a rigidity property for the homotopy invariant stable linear framed presheaves is established as a consequence a variant of gabber rigidity theorem is obtained for a cohomology theory representable in the motivic stable homotopy category by a torsion spectrum with gw k of rank coprime to the exponential characteristic of the base field k it is shown that the values of such cohomology theories at an essentially smooth henselian ring and its residue field coincide the result is applicable to cohomology theories representable by n torsion spectra as well as to the ones representable by periodic spectra and spectra related to witt groups

rigidity for henselian local rings and mathbb a 1 representable theories we prove that for a large class of mathbb a 1 representable theories including all orientable theories it is possible to construct transfer maps and to prove rigidity theorems similar to those of gabber for algebraic k theory this extends rigidity results of panin and yagunov from algebraically closed fields to arbitrary infinite ones

the construction of weight two arithmetic cohomology let x be a regular noetherian scheme in a letter to soul6 be beilinson conjectured for each non negative integer n the existence of complexes f n x of zariski sheaves on x whose hypercohomology should be in the same relation to algebraic k theory as classical cohomology is to topological ktheory and proposed certain properties that these complexes should satisfy this investigation was continued in l where the conjectured complexes were moved to the a at other points we work only up to 2torsion or up to torsion involving primes dividing the residue field characteristics there seems good reason to expect our complex f 2 x to in fact exactly satisfy all the axioms we now sketch the construction of f 2 x and recall the axioms let a be a regular noetherian ring let w s p e c a t z s p e c a t t t 1 let b bl b 2 b be a finite sequence of exceptional units of a i e b i and

third homology of general linear groups over rings with many units for a commutative ring r with many units we describe the kernel of h3 inc h3 gl2 r z h3 gl3 r z moreover we show that the elements of this kernel are of order at most two as an application we study the indecomposable part of k3 r

a look ahead lanczos algorithm for unsymmetric matrices abstract the two sided lanczos algorithm is known to suffer from serious breakdowns these occur when the associated moment matrix does not permit triangular factorization we modify the algorithm slightly so that it corresponds to using a 2 x 2 pivot in triangular factorization whenever a 1 x 1 pivot would be dangerous the incidence of breakdown is greatly reduced the price paid is that the tridiagonal matrix produced by the algorithm now has bumps whenever a 2 x 2 pivot is used author

a convergence analysis for nonsymmetric lanczos algorithms a convergence analysis for the nonsymmetric lanczos algorithm is presented by using a tridiagonal structure of the algorithm some identities concerning ritz values and ritz vectors are established and used to derive approximation bounds in particular the analysis implies the classical results for the symmetric lanczos algorithm

motivic spheres and the image of the suslin hurewicz map we show that an old conjecture of a a suslin characterizing the image of a hurewicz map from quillen k theory in degree n to milnor k theory in degree n admits an interpretation in terms of unstable mathbb a 1 homotopy sheaves of the general linear group using this identification we establish suslin s conjecture in degree 5 for infinite fields having characteristic unequal to 2 or 3 we do this by linking the relevant unstable mathbb a 1 homotopy sheaf of the general linear group to the stable mathbb a 1 homotopy of motivic spheres

on the k theory of local fields in this paper we establish a connection between the quillen k theory of certain local fields and the de rham witt complex of their rings of integers with logarithmic poles at the maximal ideal the fieldsk we consider are complete discrete valuation fields of characteristic zero with perfect residue field k of characteristic p 2 when k contains the pth roots of unity the relationship between the k theory with z p coefficients and the de rham witt complex can be described by a sequence

a bloch wigner exact sequence over local rings abstract in this article we extend the bloch wigner exact sequence over local rings where their residue fields have more than nine elements moreover we prove van der kallen s theorem on the presentation of the second k group of local rings such that their residue fields have more than four elements note that van der kallen proved this result when the residue fields have more than five elements although we prove our results over local rings all our proofs also work over semilocal rings where all their residue fields have similar properties as the residue field of local rings

rigidity in motivic homotopy theory we show that extensions of algebraically closed fields induce full and faithful functors between the respective motivic stable homotopy categories with finite coefficients

gersten conjecture for equivariant k theory and applications for a reductive group scheme g over a regular semi local ring a we prove the gersten conjecture for the equivariant k theory as a consequence we show that if f is the field of fractions of a then k g 0 a cong k g 0 f generalizing the analogous result for a dvr by serre inst hautes etudes sci publ math 34 37 52 1968 we also show the rigidity for the k theory with finite coefficients of a henselian local ring in the equivariant setting we use this rigidity theorem to compute the equivariant k theory of algebraically closed fields

the gersten conjecture for milnor k theory we prove that the n th milnor k group of an essentially smooth local ring over an infinite field coincides with the n n motivic cohomology of the ring this implies levine s generalized bloch kato conjecture

robust stabilization of uncertain linear systems quadratic stabilizability and h sup infinity control theory the problem of robustly stabilizing a linear uncertain system is considered with emphasis on the interplay between the time domain results on the quadratic stabilization of uncertain systems and the frequency domain results on h sup infinity optimization a complete solution to a certain quadratic stabilization problem in which uncertainty enters both the state and the input matrices of the system is given relations between these robust stabilization problems and h sup infinity control theory are explored it is also shown that in a number of cases if a robust stabilization problem can be solved via lyapunov methods then it can be also be solved via h sup infinity control theory based methods

reduced k theory of azumaya algebras abstract in the theory of central simple algebras often we are dealing with abelian groups which arise from the kernel or co kernel of functors which respect transfer maps for example k functors since a central simple algebra splits and the functors above are trivial in the split case one can prove certain calculus on these functors the common examples are kernel or co kernel of the maps k i f k i d where k i are quillen k groups d is a division algebra and f its center or the homotopy fiber arising from the long exact sequence of above map or the reduced whitehead group sk 1 in this note we introduce an abstract functor over the category of azumaya algebras which covers all the functors mentioned above and prove the usual calculus for it this for example immediately shows that k theory of an azumaya algebra over a local ring is almost the same as k theory of the base ring the main result is to prove that reduced k theory of an azumaya algebra over a henselian ring coincides with reduced k theory of its residue central simple algebra the note ends with some calculation trying to determine the homotopy fibers mentioned above

homology of classical lie groups made discrete ii h2 h3 and relations with scissors congruences the present work extends a number of earlier results see s 7 15 181 for example the following fundamental exact sequence essentially due to bloch and wigner in somewhat different form but not published by them can be found in dupont and sah 7 o o z th3 sl 2 n a a o 2 12 the group y c is defined in 1 4 1 8 and is isomorphic to the group pc considered in dupont and sah 7 also the study of the scissors congruence problem had led to two exact sequences in dupont s theorems 1 3 and 1 43 that roughly correspond to the f eigenspaces of 2 12 under the action of complex conjugation o a h su 2 s3 l wo w z h su 2 to 0 1

bloch wigner theorem over rings with many units the bloch wigner theorem over an algebraically closed field f determines a connection between the third homology of sl2 f and the group describing the nontrivial relations on the tensors of the form x otimes 1 x x in f times backslash 1 the purpose of this article is to provide a version of this theorem over rings with large number of unites for examples semi local rings with infinite residue fields

geometric models for higher grothendieck witt groups in mathbb a 1 homotopy theory we show that the higher grothendieck witt groups a k a algebraic hermitian k groups are represented by an infinite orthogonal grassmannian in the mathbb a 1 homotopy category of smooth schemes over a regular base for which 2 is a unit in the ring of regular functions we also give geometric models for various mathbb p 1 and s 1 loop spaces of hermitian k theory

a riccati equation approach to the stabilization of uncertain linear systems this paper presents a method for designing a feedback control law to stabilize a class of uncertain linear systems the systems under consideration contain uncertain parameters whose values are known only to within a given compact bounding set furthermore these uncertain parameters may be time varying the method used to establish asymptotic stability of the closed loop system obtained when the feedback control is applied involves the use of a quadratic lyapunov function the main contribution of this paper involves the development of a computationally feasible algorithm for the construction of a suitable quadratic lyapunov function once the lyapunov function has been obtained it is used to construct the stabilizing feedback control law the fundamental idea behind the algorithm presented involves constructing an upper bound for the lyapunov derivative corresponding to the closed loop system this upper bound is a quadratic form by using this upper bounding procedure a suitable lyapunov function can be found by solving a certain matrix riccati equation another major contribution of this paper is the identification of classes of systems for which the success of the algorithm is both necessary and sufficient for the existence of a suitable quadratic lyapunov function

system order reduction in robust stabilization problems robust stabilization via state feedback is considered using a fixed quadratic lyapunov function approach quadratic stabilization we investigate the possibility of reducing the quadratic stabilization problem of a given uncertain system to a similar problem for an uncertain subsystem with a fewer number of states this subsystem is the so called regular subsystem associated with the original system it is shown that when some of the control input channels of the given uncertain system are free of uncertainty this reduction is possible we show that a given uncertain system is quadratically stabilizable via linear state feedback if and only if the same holds for its regular subsystem when the regular subsystem is quadratically stabilizable via linear state feedback a simple formula for a controller that quadratically stabilizes the original system is given we also present an example of an uncertain system that is not quadratically stabilizable even though its regular subsystem can be quadratically

an explicit formula for lanczos polynomials abstract the lanczos algorithm for tridiagonalizing a given matrix a generates a sequence of approximating matrices a p that can naturally be obtained as restrictions to subspaces the eigenvalues of these approximating matrices are well known to be good approximations to the extreme eigenvalues of a in this paper we produce explicit formulas for the characteristics polynomials of the a p in terms of the eigenvalues of a these formulas can be used to explain heuristically why these approximations are often quite good at present we have no concrete analytic argument that explains the quality of the approximation the main result draws on the formal relationship between the lanczos algorithm and pade approximations to the moment generating function of a this result is one of the few analytic results available for the unsymmetric lanczos algorithm

third homology of general linear groups the third homology group of gln r is studied where r is a ring with many units with center z r the main theorem states that if k1 z r q k1 r q e g r a commutative ring or a central simple algebra then h3 gl2 r q h3 gl3 r q is injective if r is commutative q can be replaced by a field k such that 1 2 k for an infinite field r resp an infinite field r such that r r 2 we get the better result that h3 gl2 r z 12 h3 gl3 r z 12 resp h3 gl2 r z h3 gl3 r z is injective as an application we study the third homology group of sl2 r and the indecomposable part of k3 r