approximation numbers of weighted composition operators we study the approximation numbers of weighted composition operators f mapsto w cdot f circ varphi on the hardy space h 2 on the unit disc for general classes of such operators upper and lower bounds on their approximation numbers are derived for the special class of weighted lens map composition operators with specific weights we show how much the weight w can improve the decay rate of the approximation numbers and give sharp upper and lower bounds these examples are motivated from applications to the analysis of relative commutants of special inclusions of von neumann algebras appearing in quantum field theory borchers triples

galois objects with normal bases for free hopf algebras of prime degree suppose r is a commutative ring and h a commutative cocommutative hopf algebra which is a free r module of prime rank the set of isomorphism classes of r algebras which are galois h objects forms an abelian group see chase and sweedler a in this paper we show that when h is an algebra over a special subring of the p adic integers containing a full set of p 1 st roots of unity then the subring consisting of those galois objects which are isomorphic to h as h module h hom h r is isomorphic to ue r u r where the notation u r means units of r congruent to 1 modx and 5 is a p 1 st root of an element b of r which will be seen to characterize the hopf algebra h the paper is organised as follows we briefly review notation related to hopf algebras integrals and galois objects in section 0 in section 1 we present the description of hopf algebras of order p given by tate and oort in their paper group schemes of prime order s such a hopf algebra h is associated with a pair of constants a b in fact hz r x x ax and h x r x xp bx where ab wi some unit of r in section 2 we describe the h module structure of h and the h module structure of a galois h object s and we state what it means for s to have a normal basis s z h as h module in section 3 we assume r contains a p l st root of 6 we show rz pz c h from which it follows that s contains a kummer order c modp s with s r and sis si i

galois module structure of the square root of the inverse different in even degree tame extensions of number fields abstract let g be a finite group and let n e be a tamely ramified g galois extension of number fields whose inverse different c n e is a square let a n e denote the square root of c n e then a n e is a locally free z g module which is in fact free provided n e has odd order as shown by erez using m taylor s theorem we can rephrase this result by saying that when n e has odd degree the classes of a n e and o n the ring of integers of n in cl z g are equal and in fact both trivial we show that the above equality of classes still holds when n e has even order assuming that n e is locally abelian this result is obtained through the study of the frohlich representatives of the classes of some torsion modules which are independently introduced in the setting of cyclotomic number fields jacobi sums together with the hasse davenport formula are involved in this study finally when g is the binary tetrahedral group we use our result in conjunction with taylor s theorem to exhibit a tame g galois extension whose square root of the inverse different has nontrivial class in cl z g

hopf galois kummer theory of formal groups let f g be an isogeny between finite n dimensional formal groups defined over r the valuation ring of some field extension k of qp let h be the hopf algebra which arises from this isogeny for such h we classify gal h the group of 7 galois objects let ai be the maximal ideal of r and let p f k denote the n tuples of m under the group operation induced by f our main result is the construction of an isomorphism from the cokernel of p f to gal where p f is the induced map from p f k to p g k in geometric language gal describes the group of isomorphism classes of principal homogeneous spaces for spec over spec geometric methods have been used by mazur to establish the above isomorphism but the proof is nonconstructive a geometric approach has also provided a formula for the cardinality of gal we give an alternative derivation of this result using formal group techniques introduction given a field k and a finite abelian group g a question in classical field theory is to determine gal k g the isomorphism classes of galois field extensions of k with group g when k has characteristic zero and contains a primitive nth root of unity classical kummer theory asserts that gal k cn u k u k n where u k denotes the units of k and cn is the cyclic group of order n furthermore there is an explicit isomorphism from u k u k n to gal cn given by sending u g u k mod u k n to k x xn u in their development of hopf galois theory chase and sweedier 1 provide criteria for an action or co action of a cocommutative hopf algebra on an exten sion of rings to be galois an algebra s which admits a galois co action by an hopf algebra h is called an galois object given a ring r and a cocommu tative hopf algebra a basic question in this theory is to determine gal the group of isomorphism classes of galois objects when k is a field and h is the dual of the group ring kg g a finite group gal becomes gal k g determining gal is also of great interest because in geometric language this describes the group of isomorphism classes of principal homogeneous spaces for spec over spec we make the conventions that all rings are commutative all formal groups are commutative and all hopf algebras are abelian commutative and cocommu tative an hopf algebra is called finite if it is finitely generated and projective as an module

tamagawa numbers for motives with noncommutative coefficients ii we provide corroborative evidence for the equivariant tamagawa number conjecture which was formulated in the first part of this article

the root number class and chinburg s second invariant let n k be a finite normal extension of number fields with g gal n k by e noether s theorem the ring of integers on of n is a projective z g module if and only if n k is at most tamely ramified thus for such n k 0 defines a class q n k on k q z g in the class group cl z g the finite torsion subgroup of the grothendieck group k z g of finitely generated g modules of finite projective dimension in 14 m taylor proved that in this case q n k is equal to the root number class wnik which was defined by ph cassou nogds and friihlich by means of the artin root numbers of the irreducible symplectic representations of g cf s chap 1 or 3 p 183 in 4 t chinburg defined galois invariants q n k i i 1 2 3 of n k in cl z g and proved q n k 2 qjn k for arbitrary n k which are at most tamely ramified since both classes q n k 2 and w are defined for all n k including those which are wildly ramified one may be led to the following conjecture

the pseudoflow algorithm a new algorithm for the maximum flow problem we introduce the pseudoflow algorithm for the maximum flow problem that employs only pseudoflows and does not generate flows explicitly the algorithm solves directly a problem equivalent to the minimum cut problem the maximum blocking cut problem once the maximum blocking cut solution is available the additional complexity required to find the respective maximum flow is o m log n a variant of the algorithm is a new parametric maximum flow algorithm generating all breakpoints in the same complexity required to solve the constant capacities maximum flow problem the pseudoflow algorithm has also a simplex variant pseudoflow simplex that can be implemented to solve the maximum flow problem one feature of the pseudoflow algorithm is that it can initialize with any pseudoflow this feature allows it to reach an optimal solution quickly when the initial pseudoflow is close to an optimal solution the complexities of the pseudoflow algorithm the pseudoflow simplex and the parametric variants of pseudoflow and pseudoflow simplex algorithms are all o mn log n on a graph with n nodes and m arcs therefore the pseudoflow simplex algorithm is the fastest simplex algorithm known for the parametric maximum flow problem the pseudoflow algorithm is also shown to solve the maximum flow problem on s t tree networks in linear time where s t tree networks are formed by joining a forest of capacitated arcs with nodes s and t adjacent to any subset of the nodes

spectra of weighted composition operators with automorphic symbols abstract let be an automorphism of the open unit disc d for such we investigate the spectra of invertible weighted composition operators u c acting on a wide class of analytic function spaces this class contains for example hardy spaces h p d weighted bergman spaces a p d and weighted banach spaces of h type we present new techniques for deducing the spectrum and for calculating the spectral radius of u c we also characterize the fredholmness of weighted composition operators on h p d and a p d

singular values and schmidt pairs of composition operators on the hardy space abstract we find the singular values and corresponding schmidt pairs of a compact composition operator c induced by z a z b where a b 1 on the classical hardy space we do so by solving a functional equation that is a generalization of schroder s equation find a function f holomorphic on the open unit disc and a complex number such that g z f z f z where is a holomorphic self map of the open unit disc with an interior fixed point and g is a bounded holomorphic function on the open unit disc in addition we find the spectrum of the weighted composition operator m g c

construction of quantum field theories with factorizing s matrices the subject of this thesis is a novel construction method for interacting relativistic quantum field theories on two dimensional minkowski space employing the algebraic framework of quantum field theory it is shown under which conditions an algebra of observables localized in a wedge shaped region of spacetime can be used to construct model theories a crucial input in this context is the modular nuclearity condition for wedge algebras which implies the existence of local observables as an application of the new method a rigorous construction of a large family of models with factorizing s matrices is obtained in an inverse scattering approach a given factorizing scattering operator is used to define certain semi localized wightman fields associated to it with the help of these fields a wedge algebra can be defined which determines the local observable content of a well defined quantum field theory in this approach the modular nuclearity condition translates to certain analyticity and boundedness conditions on the formfactors of wedge local observables these conditions are shown to hold for a large class of underlying s matrices including the scattering operators of the sinh gordon model and the scaling ising model as special examples the so constructed models are investigated with respect to their scattering properties they are shown to solve the inverse scattering problem for the underlying s matrices and a proof of asymptotic completeness for these models is given

some examples of composition operators and their approximation numbers on the hardy space of the bi disk we give examples of composition operators c phi on h 2 d 2 showing that the condition phi infty 1 is not sufficient for their approximation numbers a n c phi to satisfy lim n to infty a n c phi 1 sqrt n 1 contrary to the 1 dimensional case we also give a situation where this implication holds we make a link with the monge amp ere capacity of the image of phi

the galois module structure of certain arithmetic principal homogeneous spaces in this paper we obtain galois module results for rings of integers of certain abelian extensions of cyclotomic extensions and of division fields for elliptic curves with complex multiplication in the cyclotomic case we consider the rings of integers of non ramified extensions and in the elliptic case we are concerned with extensions arising from the selmer group of the curve in each case we can describe those rings of integers which are free galois modules by means of l function congruences this then is a further instance of how galois module structure is dominated by l functions with the classic case being the tame theory see f t3 where it is the root numbers of the sympletic characters of the galois group which determine the galois module structure in broad terms the results of this paper arise from marrying the algebraic results in tl and certain powerful results from iwasawa theory both cyclotomic and elliptic for aesthetic reasons as well as for the sake of brevity we introduce notation which will simultaneously cover both the cyclotomic and the elliptic case we begin by defining the notation for

a new algorithm for the open pit mine production scheduling problem for the purpose of production scheduling open pit mines are discretized into three dimensional arrays known as block models production scheduling consists of deciding which blocks should be extracted when they should be extracted and what to do with the blocks once they are extracted blocks that are close to the surface should be extracted first and capacity constraints limit the production in each time period since the 1960s it has been known that this problem can be cast as an integer programming model however the large size of some real instances 3 10 million blocks 15 20 time periods has made these models impractical for use in real planning applications thus leading to the use of numerous heuristic methods in this article we study a well known integer programming formulation of the problem that we refer to as c pit we propose a new decomposition method for solving the linear programming relaxation lp of c pit when there is a single capacity constraint per time period this algorithm is based on exploiting the structure of the precedence constrained knapsack problem and runs in o mn log n in which n is the number of blocks and m a function of the precedence relationships in the mine our computations show that we can solve in minutes the lp relaxation of real sized mine planning applications with up to five million blocks and 20 time periods combining this with a quick rounding algorithm based on topological sorting we obtain integer feasible solutions to the more general problem where multiple capacity constraints per time period are considered our implementation obtains solutions within 6 of optimality in seconds a second heuristic step based on local search allows us to find solutions within 3 in one hour on all instances considered for most instances we obtain solutions within 1 2 of optimality if we let this heuristic run longer previous methods have been able to tackle only instances with up to 150 000 blocks and 15 time periods

nuclear maps and modular structures ii applications to quantum field theory a correspondence between spectral properties of modular operators appearing in quantum field theory and the hamiltonian is established it allows to prove the distal split property for a wide class of models conversely any model having this property is shown to satisfy the haag swieca compactness criterion the results lead to a new type of nuclearity condition which can be applied to quantum field theories on arbitrary space time manifolds

a computational study of the pseudoflow and push relabel algorithms for the maximum flow problem we present the results of a computational investigation of the pseudoflow and push relabel algorithms for the maximum flow and minimum s t cut problems the two algorithms were tested on several problem instances from the literature our results show that our implementation of the pseudoflow algorithm is faster than the best known implementation of push relabel on most of the problem instances within our computational study

hardy spaces for the strip abstract in this paper we shall study hardy spaces of analytic functions in a strip s our main result is on one hand an intrinsic characterization of the spaces and on the second that polynomials are dense we also present an orthogonal in h 2 s basis of polynomials

nuclear maps and modular structures i general properties abstract we establish a correspondence between the split property of inclusions a b of von neumann algebras and nuclearity properties of the natural embeddings p l a lp b p 1 2 given by the modular theory

relative galois module structure and steinitz classes of dihedral extensions of degree 8 abstract let k be a number field and ok its ring of integers let be the dihedral group of order 8 let m be a maximal ok order in k containing ok and c l m its class group we denote by r m the set of realizable classes that is the set of classes c c l m such that there exists a galois extension n k at most tamely ramified with galois group isomorphic to and the class of m ok on equal to c where on is the ring of integers of n in this article we prove that r m is a subgroup of c l m provided that k and the fourth cyclotomic field of q are linearly disjoint and the class number of k is odd to this end we will solve an embedding problem connected with steinitz classes of galois extensions in addition for any k with odd class number we show that the set of steinitz classes of tame dihedral extension of k is the full class group of k

tameness and local normal bases for objects of finite hopf algebras let r be a commutative ring s an r algebra h a hopf ralgebra both finitely generated and projective as r modules and suppose s is an h object so that h homr h r acts on s via a measuring let i be the space of left integrals of h we say s has normal basis if s h as h modules and s has local normal bases if sp hp as hp modules for all prime ideals p of r when r is a perfect field h is commutative and cocommutative and certain obvious necessary conditions on s hold then s has normal basis if and only if is r sh if r is a domain with quotient field k h is cocommutative and l s or k has normal basis as h 0 k module then s has local normal bases if and only if is r sh suppose k is a number field with ring of integers r l is a finite galois extension of k with galois group g and s is the integral closure of r in l then g acts as a group of automorphisms of s relative galois module theory seeks to understand s as an rg module via this action the most basic question is to inquire whether s is locally isomorphic to rg that is for all primes p of r sp has a local normal basis as a free rp module this question was answered by emmy noether who showed that s has a local normal basis at every prime p of r if and only if l k is tamely ramified here tamely ramified means that at each prime ideal p of r the ramification index of any prime p of s lying over p is relatively prime to the residue field characteristic it is well known that this latter condition is equivalent to the surjectivity of the trace map tr s r tr s ezacg s the purpose of this paper is to formulate and prove an extension of noether s theorem to objects of finite cocommutative hopf algebras assume now only that r is a commutative ring h is a hopf r algebra which is a finitely generated projective r module and h homr h r the dual hopf algebra a commutative r algebra s is an h object if s is a right h comodule via a map a s s 0 h which is an r algebra homomorphism if s is an h object then a induces a measuring a h s s in the sense of sweedler 21 galois module theory in this setting is the study of s as an h module via the measuring a the trivial example is s h itself with a a the comultiplication on h so we say that the h object has normal basis if s h as h modules and s has local normal bases if for all primes p of r sp hp as h modules since rgf rg as rg modules for g any finite group this notion of normal basis when specialized to h rg is equivalent to the classical notion received by the editors may 7 1985 and in revised form january 7 1986 1980 mathematics subject classification 1985 revision primary 13b05 secondary 16a24 1986 american mathematical society 0002 9947 86 1 00 25 per page

hardy spaces that support no compact composition operators we consider for g a simply connected domain and 0 p the hardy space hp g formed by fixing a riemann map of the unit disc onto g and demanding of functions f holomorphic on g that the integrals of f p over the curves z r be bounded for 0 r 1 the resulting space is usually not the one obtained from the classical hardy space of the unit disc by conformal mapping this is reflected in our main theorem hp g supports compact composition operators if and only if g has finite one dimensional hausdorff measure our work is inspired by an earlier result of matache proc amer math soc 127 1999 1483 who showed that the hp spaces of half planes support no compact composition operators our methods provide a lower bound for the essential spectral radius which shows that the same result holds with compact replaced by riesz we prove similar results for bergman spaces with the hardy space condition g has finite hausdorff 1 measure replaced by g has finite area finally we characterize those domains g for which every composition operator on either the hardy or the bergman spaces is bounded

on frohlich s conjecture for rings of integers of tame extensions where is the complex conjugate of z the constant w z is called the artin root number of in m l j martinet showed that om is not always a free zf module when n q and when f is a quaternion group of order 8 subsequently in f5 a frtihlich showed that the class of u in this special case is determined by the sign of the artin root number of the irreducible sympletic character of 2 we now return to the general case and let ci tif be the class group of locally free zf modules given a locally free 2 f module x we denote its class by x philippe cassou nogu6s defined a class t w in ci f which is defined solely in terms of the values of artin root numbers of sympletic characters this class will be defined in w after numerous computations by fr6hlich et al cf f6 c t3 for instance fr6hlich conjectured the result given below which we prove in this paper