cyclic homology comodules and mixed complexes relating cyclic and hochschild homology one can see that any formula for hc a a will not only involve hc a and hc a but also the periodicity operator s and the hochschild groups fortunately if the map s brings in some complications it also endows hc a with a comodule structure over the cyclic homology of the ground ring k actually the comodule structure exists already on the complex level a convenient complex on which s has the form of a canonical surjection is connes s double complex with differentials b and b now the idea is to view the cyclic homology of an algebra as a composite functor

on the negative k theory of schemes in finite characteristic abstract we show that if x is a d dimensional scheme of finite type over an infinite perfect field k of characteristic p 0 then k i x 0 and x is k i regular for i d 2 whenever the resolution of singularities holds over k this proves the k dimension conjecture of weibel c weibel k theory and analytic isomorphisms invent math 61 1980 177 197 2 9 except for d 1 i d 2 in all characteristics assuming the resolution of singularities

on the general linear group and hochschild homology our main result here is a rational computation of the homology of the adjoint action of the infinite general linear group of an arbitrary ring before stating the result we establish some notation and conventions rings are associative and with unit if a is a ring then gl a uk 0glk a is its infinite general linear group an a bimodule is an abelian group b which is both a left a module and a right a module and satisfies alb a2 al ba2 for ai ea be b for example b a if b is an a bimodule then m b uk omk b is the infinite additive group of matrices with entries in b conjugation defines an action the adjoint action of gl a on m b note that an a q bimodule is just an a bimodule which is also a rational vector space if b is an a q bimodule then hj a c q b denotes the hochschild homology of a q with coefficients in b our main result it appears in slightly more detailed form as theorem v 3 is

conway spheres as ideal points of the character variety we prove that any bonahon siebenmann family of conway spheres for a hyperbolic link is associated to an ideal point of the character variety of the link

on a conjecture of vorst let k be an infinite perfect field of positive characteristic such that strong resolution of singularities holds over k we prove that a localization of a d dimensional commutative k algebra r of finite type is kd 1 regular if and only if it is regular this partially affirms a conjecture of vorst

on culler shalen seminorms and dehn filling if f is a finitely generated discrete group and g a complex algebraic lie group the g character variety of r is an affine algebraic variety whose points correspond to characters of representations of r with values in g marc culler and peter shalen developed the theory of sl2 c character varieties of finitely generated groups and applied their results to study the topology of 3 dimensional manifolds in the papers 6 7 8 consider the exterior m of a hyperbolic knot lying in a closed connected orientable 3 manifold the mostow rigidity theorem implies that the holonomy representation p iri m isom h3 psl2 c is unique up to conjugation and taking complex conjugates the orientability of m can be used to show 1 lifts to a representation p e hom ri m sl2 c whose character determines an essentially unique point of xp of x iri m the sl2 c character variety of irl m culler and shalen 8 proved that the component x0 of x x1ri m which contains x is a curve one of their major contributions was to show how x0 determines a norm on h1 wm r which encodes many topological properties of m in particular it provides information on the dehn fillings of m their construction may be applied to arbitrary curves in the sl2 c character variety of a connected compact orientable irreducible 3manifold whose boundary is a torus though in this generality one can only guarantee that it will define a seminorm the first half of this paper is devoted to the development of the general theory of culler shalen seminorms defined for curves of psl2 c characters by working over psl2 c we obtain a theory that is more generally applicable than its sl2 c counterpart while being only mildly more difficult to set up in the second half of this paper we apply the theory of culler shalen seminorms to study the dehn filling operation in particular we examine the relationship between fillings which yield manifolds having a positive dimen

the obstruction to excision in k theory and in cyclic homology let f a b be a ring homomorphism of not necessarily unital rings and i triangleleft a an ideal which is mapped by f isomorphically to an ideal of b the obstruction to excision in k theory is the failure of the map between relative k groups k a i k b f i to be an isomorphism it is measured by the birelative groups k a b i similarly the groups hn a b i measure the obstruction to excision in negative cyclic homology we show that the rational jones goodwillie chern character induces an isomorphism ch k a b i otimes mathbb q overset sim to hn a otimes mathbb q b otimes mathbb q i otimes mathbb q

bayesian bounds for matched field parameter estimation matched field methods concern estimation of source locations and or ocean environmental parameters by exploiting full wave modeling of acoustic waveguide propagation because of the nonlinear parameter dependence of the signal field the estimate is subject to ambiguities and the sidelobe contribution often dominates the estimation error below a threshold signal to noise ratio snr to study the matched field performance three bayesian lower bounds on mean square error are developed the bayesian crame spl acute r rao bound bcrb the weiss weinstein bound wwb and the ziv zakai bound zzb particularly for a multiple frequency multiple snapshot random signal model a closed form minimum probability of error associated with the likelihood ratio test is derived which facilitates error analysis in a wide scope of applications analysis and example simulations demonstrate that 1 unlike the local crb the bcrb is not achieved by the maximum likelihood estimate mle even at high snr if the local performance is not uniform across the prior parameter space 2 the zzb gives the closest mle performance prediction at most snr levels of practical interest 3 the zzb can also be used to determine the necessary number of independent snapshots achieving the asymptotic performance of the mle at a given snr 4 incoherent frequency averaging which is a popular multitone processing approach reduces the peak sidelobe error but may not improve the overall performance due to the increased ambiguity baseline and finally 5 effects of adding additional parameters e g environmental uncertainty can be well predicted from the parameter coupling

cyclic homology cdh cohomology and negative k theory we prove a blow up formula for cyclic homology which we use to show that infinitesimal k theory satisfies cdh descent combining that result with some computations of the cdh cohomology of the sheaf of regular functions we verify a conjecture of weibel predicting the vanishing of algebraic k theory of a scheme in degrees less than minus the dimension of the scheme for schemes essentially of finite type over a field of characteristic zero

information theoretic analysis of iterated bayesian acoustic source localization in a static ocean waveguide fundamental constructs of information theory are applied to quantify the performance of iterated sequential bayesian localization of a time harmonic source in a range and time invariant acoustic waveguide using the segmented fourier transforms of the received pressure time series the nonlinear relation defined by acoustic propagation between the source location and the received narrowband spectral components is treated as a nonlinear communication channel the performance analysis includes mismatch between the acoustic channel and the model channel on which the bayesian inference is based source location uncertainty is quantified by the posterior probability density of source location by the posterior entropy and associated uncertainty area by the information gain relative entropy at each iteration and by large ensemble limits of these quantities a computational example for a vertical receiver array in a shallow water waveguide is presented with acoustic propagation represented by normal mode

explicit ziv zakai lower bound for bearing estimation the extended ziv zakai bound for vector parameters is used to develop a lower bound on the mean square error in estimating the 2 d bearing of a narrowband planewave signal using planar arrays of arbitrary geometry the bound has a simple closed form expression that is a function of the signal wavelength the signal to noise ratio snr the number of data snapshots the number of sensors in the array and the array configuration analysis of the bound suggests that there are several regions of operation and expressions for the thresholds separating the regions are provided in the asymptotic region where the number of snapshots and or snr are large estimation errors are small and the bound approaches the inverse fisher information this is the same as the asymptotic performance predicted by the local cramer rao bound for each value of bearing in the a priori performance region where the number of snapshots or snr is small estimation errors are distributed throughout the a priori parameter space and the bound approaches the a priori covariance in the transition region both small and large errors occur and the bound varies smoothly between the two extremes simulations of the maximum likelihood estimator mle demonstrate that the bound closely predicts the performance of the mle in all regions

non standard components of the character variety for a family of montesinos knots we show that there are montesinos knots with n 1 geq 4 tangles whose character varieties contain arbitrarily many irreducible components of dimension d for any 1 le d le n 2 moreover these irreducible components can be chosen so that the trace of the meridian is non constant

mean squared error and threshold snr prediction of maximum likelihood signal parameter estimation with estimated colored noise covariances an interval error based method mie of predicting mean squared error mse performance of maximum likelihood estimators mles is extended to the case of signal parameter estimation requiring intermediate estimation of an unknown colored noise covariance matrix an intermediate step central to adaptive array detection and parameter estimation the successful application of mie requires good approximations of two quantities 1 interval error probabilities and 2 asymptotic snrrarrinfin local mse performance of the mle exact general expressions for the pairwise error probabilities that include the effects of signal model mismatch are derived herein that in conjunction with the union bound provide accurate prediction of the required interval error probabilities the crameacuter rao bound crb often provides adequate prediction of the asymptotic local mse performance of mle the signal parameters however are decoupled from the colored noise parameters in the fisher information matrix for the deterministic signal model rendering the crb incapable of reflecting loss due to colored noise covariance estimation a new modification of the crb involving a complex central beta random variable different from but analogous to the reed mallett and brennan beta loss factor provides a working solution to this problem facilitating mse prediction well into the threshold region with remarkable accuracy

bredon style homology cohomology and riemann roch for algebraic stacks one of the main obstacles for proving riemann roch for algebraic stacks is the lack of cohomology and homology theories that are closer to the k theory and g theory of algebraic stacks than the traditional cohomology and homology theories for algebraic stacks in this paper we study in detail a family of cohomology and homology theories which we call bredon style theories that are of this type and in the spirit of the classical bredon cohomology and homology theories defined for the actions of compact topological groups on topological spaces we establish riemann roch theorems in this setting it is shown elsewhere that such riemann roch theorems provide a powerful tool for deriving formulae involving virtual fundamental classes associated to dg stacks for example moduli stacks of stable curves provided with a virtual structure sheaf associated to a perfect obstruction theory we conclude the present paper with a brief application of this nature

on the vanishing of negative k groups we show that for a d dimensional scheme x essentially of finite type over an infinite perfect field k of characteristic p 0 the negative k groups k q x vanish for q d provided that strong resolution of singularities holds over the field k

toric varieties monoid schemes and cdh descent we give conditions for the mayer vietoris property to hold for the algebraic k theory of blow up squares of toric varieties and schemes using the theory of monoid schemes these conditions are used to relate algebraic k theory to topological cyclic homology in characteristic p to achieve our goals we develop many notions for monoid schemes based on classical algebraic geometry such as separated and proper maps and resolution of singularities

rate distortion bounds on passive sonar performance information theory provides a novel perspective on passive sonar performance analysis this approach begins by partitioning the search space and then considers the problem of assigning a source to the correct partition based on pressure observations from a hydrophone array prior work described necessary conditions for achieving arbitrarily small probability of error pe as a tradeoff between signal to noise ratio snr and the range accuracy of the partitions this paper presents a method to extend these results using the rate distortion function to find necessary conditions for any pe the gaussian channel bound sets an upper limit on the information rate received at the array through the rate distortion function this upper limit on the information rate implies a lower bound on pe for a given partition furthermore the current work describes a tradeoff between range accuracy pe and snr examples of this tradeoff are evaluated for both one dimensional bearing and two dimensional range depth estimation problems

excision in cyclic homology and in rational algebraic k theory for a precise definition see 1 below by replacing everywhere k by k q one obtains the corresponding notion in rational algebraic k theory the above definition has an obvious counterpart for cyclic homology and algebras over a fixed field in the case of a general commutative ground ring k some restrictions on the class of allowable extensions seem inevitable due to the well known limitations of cyclic homology considered as a homology functor for algebras not flat over a ground ring an extension of k algebras will be called pure if it is pure as an extension of k modules in the sense of p m cohn 6 cf also appendix a 3 below the class of pure extensions contains e g i extensions which admit a k module splitting ii extensions a e r s with s flat over k in fact one among the several possible characterizations of purity says that an extension is pure if and only if the underlying extension of k modules is an inductive limit of split extensions see theorem a 4 of appendix a below everywhere in this paper the word excision used in the context of cyclic homology will mean excision with respect to the class of pure extensions the second purpose of the present paper is to give a complete characterization of the class of algebras possessing the excision property in cyclic homology before stating the corresponding result we need the following definition let us

bass nk groups and cdh fibrant hochschild homology the k theory of a polynomial ring r t contains the k theory of r as a summand for r commutative and containing q we describe k r t k r in terms of hochschild homology and the cohomology of kahler differentials for the cdh topology we use this to address bass question whether kn r kn r t implies kn r kn r t1 t 2 the answer to this question is affirmative when r is essentially of finite type over the complex numbers but negative in general

dehn surgery on knots in d dehn considered the following method for constructing 3 manifolds remove a solid torus neighborhood n k of some knot x in the 3 sphere s and sew it back differently in particular he showed that taking x to be the trefoil one could obtain infinitely many non simply connected homology spheres o in this way let mx s n k then the different resewings are parametrized by the isotopy class r of the simple closed curve on the torus omk that bounds a meridional disk in the re attached solid torus we denote the resulting closed oriented 3 manifold by mk and say that it is obtained by r dehn surgery on x more generally one can consider the manifolds ml obtained by r dehn surgery on a fc component link l xi u u kk in s where r r it turns out that every closed oriented 3 manifold can be constructed in this way wal lie thus a good understanding of dehn surgery might lead to progress on general questions about the structure of 3 manifolds starting with the case of knots it is natural to extend the context a little and consider the manifolds m r obtained by attaching a solid torus v to an arbitrary compact oriented irreducible every 2 sphere bounds a 3 ball 3 manifold m with dm an incompressible torus where r is the isotopy class slope on dm of the boundary of a meridional disk of v we say that m r is the result of r dehn filling on m an observed feature of this construction is that

plane curves associated to character varieties of 3 manifolds consider a compact 3 manifold m with boundary consisting of a single torus the papers cs1 cs2 cgls discuss the variety of characters of sl2 c representations of zl m and some of the ways in which the topological structure of m is reflected in the algebraic geometry of the character variety we will describe in this paper a certain affine algebraic curve dm which is naturally associated to the character variety of m a basis b 9 1 for the peripheral subgroup of m determines an embedding p8 of dm into c x c with coordinates l and m the closure in c 2 of ps dm is a plane algebraic curve and therefore is defined by a polynomial am n0 m that after certain normalizations is uniquely determined up to multiplication by constants the polynomial am b is effectively computable and is an invariant of the manifold m together with the choice of basis b the results in this paper describe how geometric properties of the character variety and hence topological properties of m are reflected by the polynomial am b in the case that m is the complement of a knot k in a homology 3 sphere we may take the basis b to consist of the longitude and meridian of k with the usual orientation conventions this basis is well defined modulo the involution which inverts both the longitude and meridian it will follow from the construction of dm that if the basis b is obtained from b by inverting both generators then the regular maps pb and pb have the same image thus ar am s am b is an invariant of the knot the polynomial am b displays in a striking way information about the incompressible surfaces in m this involves the newton polygon of am b

algebraic k theory and descent for blow ups we prove that algebraic k theory satisfies pro descent for abstract blow up squares of noetherian schemes as an application we derive weibel s conjecture on the vanishing of negative k groups

fundamentals of statistical signal processing estimation theory minimum variance unbiased estimation cramer rao lower bound linear models general minimum variance unbiased estimation best linear unbiased estimators maximum likelihood estimation least squares method of moments the bayesian philosophy general bayesian estimators linear bayesian estimators kalman filters summary of estimators extension for complex data and parameters

k theory of endomorphism rings and of rings of invariants abstract this paper gives the following description ofk0of the endomorphism ring of a finitely generated projective module t heorem let t be a ring and p a finitely generated projective t module let i be the trace ideal of p then k0 end pt is isomorphic to a subgroup of k0 t i if further the natural map k1 t k1 t i is surjective then k0 end pt is isomorphic to the subgroup of k0 t generated by the direct summands of pn for n n as a corollary we can determinek0of the ring of invariants for many free linear actions in particular the following result is proved t heorem let v be a fixed point free linear representation of a finite group g over a field k of characteristic zero and let s v be the symmetric algebra of v let k be any finite dimensional k vector space then k0 s v g ks k s v g ks k similar results are given for suitable noncommutative versions ofs v

cyclic homology and equivariant homology the purpose of this paper is to explore the relationship between the cyclic homology and cohomology theories of connes 9 11 see also loday and quillen 20 and if equivariant homology and cohomology theories here ii is the circle group the most general results involve the definitions of the cyclic homology of cyclic chain complexes and the notions of cyclic and cocyclic spaces so precise statements will be postponed until w 3 in this introduction we explain some of the formal similarities between the cyclic theory and the equivariant theory and give two examples where the general results apply let a be an associative algebra over a commutative ring k then one can form the cyclic homology hc a and cohomology hc a of a these groups have periodicity operators

the k theory of toric varieties recent advances in computational techniques for k theory allow us to describe the k theory of toric varieties in terms of the k theory of fields and simple cohomological data

infinitesimal cohomology and the chern character to negative cyclic homology there is a chern character from k theory to negative cyclic homology we show that it preserves the decomposition coming from adams operations at least in characteristic zero

k regularity cdh fibrant hochschild homology and a conjecture of vorst it is a well known fact that algebraic if theory is homotopy invariant as a functor on regular schemes if x is a regular scheme then the natural map kn x kn x x 1 is an isomorphism for all n ez this is false in general for nonregular schemes and rings to express this failure bass introduced the terminology that for any contravari ant functor v defined on schemes a scheme x is called v regular if the pullback maps v x v x x ar are isomorphisms for all r 0 if x spec i we also say that r is p regular thus regular schemes are ifn regular for every n in contrast it was observed as long ago as in 2 that a nonreduced affine scheme can never be k regular in particular if a is an artinian ring that is a o dimensional noetherian ring then a is regular that is reduced if and only if a is ifi regular in 17 vorst conjectured that for an affine scheme x of finite type over a field f and of dimension d regularity and k i regularity are equivalent vorst proved this conjecture for d 1 by proving that regularity implies normality in this paper we prove vorst s conjecture in all dimensions provided the char acteristic of the ground field f is zero in fact we prove a stronger statement we say that x is regular in codimension n in x note that for all n z if a ring r is ifn regular then it is kn i regular this is proved in 17 for n 1 and in 6 4 4 for n kh x where k x is the algebraic if theory spectrum of x and kh x is the homotopy if theory of x defined in 19 we write fk r for fk spec r theorem 0 1 let r be a commutative ring which is essentially of finite type over a field f of characteristic 0 then the following hold a if tk r is n connected then r is regular in codimension khi x is an isomorphism for i n and a surjection for i n 1 so that fk x

three dimensional manifolds kleinian groups and hyperbolic geometry 1 a conjectural picture of 3 manifolds a major thrust of mathematics in the late 19th century in which poincare had a large role was the uniformization theory for riemann surfaces that every conformai structure on a closed oriented surface is represented by a riemannian metric of constant curvature for the typical case of negative euler characteristic genus greater than 1 such a metric gives a hyperbolic structure any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions the exceptional cases the sphere and the torus have spherical and euclidean structures three manifolds are greatly more complicated than surfaces and i think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 or more except perhaps for the fact that so many 3 manifolds are beautiful the situation has changed so that i feel fairly confident in proposing the

detection of essential surfaces in 3 manifolds with sl 2 trees according to bass serre theory se given a field f with a discrete valuation v f z there is a canonical way to construct a simplicial tree t tf v on which sl2 f acts simplicially without inversion when a group has a representation into the group sl2 f then there is an induced action of on the tree via the representation if further is the fundamental group of a compact 3 manifoldm and the action of on the tree t is nontrivial meaning that there is no point on t fixed by every element of then the action induces a splitting of m along an essential surface in the following way let m p m be the universal covering there is a 1 m equivariant map f m t which is transverse to the set e of midpoints of edges in t such that p f 1 e is an essential surface in m we say such an essential surface is associated to or detected by an sl2 tree a natural question is which essential surfaces in a compact 3 manifold can be associated to sl2 trees and how do they depend on the choice of the field and discrete valuation this is the main issue we are going to address in this paper any 3 manifold mentioned in this paper is automatically assumed orientable and connected by an essential surface in a compact 3 manifold m we mean an orientable properly embedded incompressible surface each component of which is neither boundary parallel in m nor bounds a 3 ball when it is a 2sphere also recall that a discrete valuation on a field f is a homomorphism v from the multiplicative group f onto the group of integers z such that v a b min v a v b for any a b f with a b 0 let v f z be a discrete valuation the set ov 0 a f v a 0 s schanuel x zhang department of mathematics suny at buffalo buffalo ny 14260 2900 usa e mail schanuel buffalo edu xinzhang math buffalo edu