construction of hyperelliptic function fields of high three rank we present several explicit constructions of hyperelliptic function fields whose jacobian or ideal class group has large 3 rank our focus is on finding examples for which the genus and the base field are as small as possible most of our methods are adapted from analogous techniques used for generating quadratic number fields whose ideal class groups have high 3 rank but one method applicable to finding large l ranks for odd primes l 3 is new and unique to function fields algorithms examples and numerical data are included

on the 3 sylow subgroup of the class group of quadratic fields in this paper we give a large number of quadratic fields q vd whose class group h d has 3 sylow subgroup h3 d with rank r3 d 1 they include 20 discriminants d 0 with r3 d 5 the associate real fields have r3 d 4 the distribution of the h3 d is studied and its relative frequency is compared with the heuristic conjectures of cohen and lenstra some of the h d are recorded since they are of special interest some related topics are also considered

circular free spectrahedra abstract this paper considers matrix convex sets invariant under several types of rotations it is known that matrix convex sets that are free semialgebraic are solution sets of linear matrix inequalities lmis they are called free spectrahedra we classify all free spectrahedra that are circular that is closed under multiplication by e i t up to unitary equivalence the coefficients of a minimal lmi defining a circular free spectrahedron have a common block decomposition in which the only nonzero blocks are on the superdiagonal a matrix convex set is called free circular if it is closed under left multiplication by unitary matrices as a consequence of a hahn banach separation theorem for free circular matrix convex sets we show the coefficients of a minimal lmi defining a free circular free spectrahedron have up to unitary equivalence a block decomposition as above with only two blocks this paper also gives a classification of those noncommutative polynomials invariant under conjugating each coordinate by a different unitary matrix up to unitary equivalence such a polynomial must be a direct sum of univariate polynomials

poisson arrivals see time averages in many stochastic models particularly in queueing theory poisson arrivals both observe see a stochastic process and interact with it in particular cases and or under restrictive assumptions it has been shown that the fraction of arrivals that see the process in some state is equal to the fraction of time the process is in that state in this paper we present a proof of this result under one basic assumption the process being observed cannot anticipate the future jumps of the poisson process

some remarks on c convexity abstract recently the study of completely positive maps has become important to the results of brown douglas and fillmore on ext a c algebra attempts to solve questions related to ext have often turned into questions about the matrix algebras mn in this paper we wish to discuss a notion of c convexity related to completely positive linear maps to state some facts about c convexity and to ask some questions about c convexity to a large degree the tone of this paper is expository

uniqueness universality and homogeneity of the noncommutative gurarij space we realize the noncommutative gurarij space ng defined by oikhberg as the fraisse limit of the class of finite dimensional 1 exact operator spaces as a consequence we deduce that the noncommutative gurarij space is unique up to completely isometric isomorphism homogeneous and universal among separable 1 exact operator spaces we also prove that ng is the unique separable nuclear operator space with the property that the canonical triple morphism from the universal tro to the triple envelope is an isomorphism we deduce from this fact that ng does not embed completely isometrically into an exact c algebra and it is not completely isometrically isomorphic to a c algebra or to a tro we also provide a canonical construction of ng which shows that the group of surjective complete isometries of ng is universal among polish groups analog results are proved in the commutative setting and more generally for m n spaces in particular we provide a new characterization and canonical construction of the gurarij banach space

randomization procedures in the computation of cumulative time distributions over discrete state markov processes this paper proposes a methodology for computing numerical bounds on cumulative time i e the time spent in a specified set of states before entering another specified set of states distributions over discrete state markov processes the methodology uses randomization procedures to compute results for appropriately defined transient markov processes the transient distribution computed is equivalent to the requisite cumulative time distribution a queueing application of the methodology to delay times in queueing networks is outlined and its efficacy is appraised by comparing the results with a sojourn time for a problem with a known distribution

the kirchberg wassermann operator system is unique in 1998 kirchberg and wassermann constructed a separable nuclear operator system with the property that the canonical unital homomorphism from the maximal c algebra onto the c envelope is injective we show that such an operator system is unique and completely order isomorphic to the operator system associated with the noncommutative poulsen simplex

compressions of compact tuples abstract we study the matrix range of a tuple of compact operators on a hilbert space and examine the notions of minimal nonsingular and fully compressed tuples in this pursuit we refine previous results by characterizing nonsingular compact tuples in terms of matrix extreme points of the matrix range further we find that a compact tuple a is fully compressed if and only if it is multiplicity free and the shilov ideal is trivial which occurs if and only if a is minimal and nonsingular fully compressed compact tuples are therefore uniquely determined up to unitary equivalence by their matrix ranges we also produce a proof of this fact which does not depend on the concept of nonsingularity

matrix convex sets without absolute extreme points abstract this article shows the existence of a class of closed bounded matrix convex sets which do not have absolute extreme points the sets we consider are noncommutative sets k x formed by taking matrix convex combinations of a single tuple x in the case that x is a tuple of compact operators with no nontrivial finite dimensional reducing subspaces k x is a closed bounded matrix convex set with no absolute extreme points a central goal in the theory of matrix convexity is to find a natural notion of an extreme point in the dimension free setting which is minimal with respect to spanning matrix extreme points are the strongest type of extreme point known to span matrix convex sets however they are not necessarily the smallest set which does so absolute extreme points a more restricted type of extreme points that are closely related to arveson s boundary enjoy a strong notion of minimality should they span this result shows that matrix convex sets may fail to be spanned by their absolute extreme points

on markov jump processes imbedded at jump epochs and their queueing theoretic applications let xt t 0 be a markov jump process and tn n 0 a subset of the jump epochs of the process xt t 0 the main results of this paper are computable formulae for the transient and invariant distributions of xtn n 0 and xtni n 0 we also characterize when the invariant distributions of xt t 0 and xtn n 0 are the same and show that in this case tn n 0 is a poisson process the results are applied to a variety of discrete slate queueing networks to obtain their state distribution as seen by customers in arrival streams departure streams and traffic streams on the arcs the main conclusion drawn is that for many traffic streams the invariant distribution seen by customers in that stream coincides with the invariant distribution of xt t 0 provided the customer in motion is excluded

quantum duality unbounded operators and inductive limits in this paper we investigate the inductive limits of quantum normed or operator spaces this construction allows us to treat the space of all noncommutative continuous functions over a quantum domain as a quantum or local operator space of all matrix continuous linear operators equipped with s quantum topology in particular we classify all quantizations of the polynormed topologies compatible with the given duality proposing a noncommutative arens mackey theorem further the inductive limits of operator spaces are used to introduce locally compact and locally trace class unbounded operators on a quantum domain and prove the dual realization theorem for an abstract quantum space

an anti pasta result for markovian systems pasta poisson arrivals see time averages is a term coined by r wolff in his well known 1982 paper in keeping with wolff s terminology we use the term anti pasta to refer to the following converse of pasta given that arrivals do indeed see time averages when must the arrival process necessarily be poisson we show that anti pasta is satisfied in a pure jump markov process provided that the arrival process corresponds to a subset of the markov process jumps

pure matrix states on operator systems abstract an operator system is a complex matricially ordered vector space that is completely order isomorphic to a unital selfadjoint subspace of a unital c algebra a matrix state on an operator system v is a unital completely positive linear map of v into a full matrix algebra pure matrix states are studied and a new and somewhat simplified proof of a krein milman type theorem of webster and winkler is given if v is 3 dimensional then the matrix state space of v is matrix affinely homeomorphic to the matricial range of some hilbert space operator with the aid of this representation pure matrix states on 3 dimensional operator systems are examined and in some cases completely determined

quantized gromov hausdorff distance abstract a quantized metric space is a matrix order unit space equipped with an operator space version of rieffel s lip norm we develop for quantized metric spaces an operator space version of quantum gromov hausdorff distance we show that two quantized metric spaces are completely isometric if and only if their quantized gromov hausdorff distance is zero we establish a completeness theorem as applications we show that a quantized metric space with 1 exact underlying matrix order unit space is a limit of matrix algebras with respect to quantized gromov hausdorff distance and that matrix algebras converge naturally to the sphere for quantized gromov hausdorff distance

matrix convexity operator analogues of the bipolar and hahn banach theorems abstract several basic results of convexity theory are generalized to the quantized matrix convex sets of wittstock these include the bipolar theorem a gauge version of the hahn banach theorem and the existence theorem for support functionals

c convexity of norm unit balls abstract we characterize the norms on b h whose unit balls are c convex we call such norms m norms and investigate their dual norms named l norms we show that the class of l norms greater than a given norm enjoys a minimum element and the class of m norms less than a given norm enjoys a maximum element these minimum and maximum elements will be determined in some cases finally we give a constructive result to obtain m norms and l norms on b h

fraisse limits in functional analysis we provide a unified approach to fraisse limits in functional analysis including the gurarij space the poulsen simplex and their noncommutative analogs we obtain in this general framework many known and new results about the gurarij space and the poulsen simplex and at the same time establish their noncommutative analogs particularly we construct noncommutative analogs of universal operators in the sense of rota

a probabilistic look at networks of quasi reversible queues probabilistic arguments are given to explain some recent results on networks of queues those results are the product form the output theorems the distributions at the jumps and the poisson character of the flows the proposed approach replaces the usual calculations by arguments showing these properties to be consequences of the behavior of the nodes in isolation

operator positivstellensatze for noncommutative polynomials positive on matrix convex sets abstract this article studies algebraic certificates of positivity for noncommutative nc operator valued polynomials on matrix convex sets such as the solution set d l called a free hilbert spectrahedron of the linear operator inequality loi l x a 0 i j 1 g a j x j 0 where a j are self adjoint linear operators on a separable hilbert space x j matrices and i is an identity matrix if a j are matrices then l x 0 is called a linear matrix inequality lmi and d l a free spectrahedron for monic lmis i e a 0 i and nc matrix valued polynomials the certificates of positivity were established by helton klep and mccullough in a series of articles with the use of the theory of complete positivity from operator algebras and classical separation arguments from real algebraic geometry since the full strength of the theory of complete positivity is not restricted to finite dimensions but works well also in the infinite dimensional setting we use it to tackle our problems first we extend the characterization of the inclusion d l 1 d l 2 from monic lmis to monic lois l 1 and l 2 as a corollary one immediately obtains the description of a polar dual of a free hilbert spectrahedron d l and its projection called a free hilbert spectrahedrop further on using this characterization in a separation argument we obtain a certificate for multivariate matrix valued nc polynomials f positive semidefinite on a free hilbert spectrahedron defined by a monic loi replacing the separation argument by an operator fejer riesz theorem enables us to extend this certificate in the univariate case to operator valued polynomials f finally focusing on the algebraic description of the equality d l 1 d l 2 we remove the assumption of boundedness from the description in the lmis case by an extended analysis however the description does not extend to lois case by counterexamples

on quadratic fields with large 3 rank davenport and heilbronn defined a bijection between classes of binary cubic forms and classes of cubic fields which has been used to tabulate the latter we give a simpler proof of their theorem then analyze and improve the table building algorithm it computes the multiplicities of the o x general cubic discriminants real or imaginary up to x in time o x and space o x or more generally in time o x x m and space o m x for a freely chosen positive m a variant computes the 3 ranks of all quadratic fields of discriminant up to x with the same time complexity but using only m o 1 units of storage as an application we obtain the first 1618 real quadratic fields with r3 4 and prove that q 5393946914743 is the smallest imaginary quadratic field with 3 rank equal to 5

local operator spaces unbounded operators and multinormed c algebras abstract in this paper we propose a representation theorem for local operator spaces which extends ruan s representation theorem for operator spaces based upon this result we introduce local operator systems which are locally convex versions of the operator systems and prove stinespring theorem for local operator systems a local operator c algebra is an example of a local operator system finally we investigate the injectivity in both local operator space and local operator system senses and prove locally convex version of the known result by choi and effros that an injective local operator system possesses unique multinormed c algebra structure with respect to the original involution and matrix topology

arveson extreme points span free spectrahedra let sm n mathbb r g denote g tuples of n times n real symmetric matrices given tuples x x 1 ldots x g in sm n 1 mathbb r g and y y 1 ldots y g in sm n 2 mathbb r g a matrix convex combination of x and y is a sum of the form begin aligned v 1 xv 1 v 2 y v 2 qquad qquad v 1 v 1 v 2 v 2 i n end aligned where v 1 mathbb r n rightarrow mathbb r n 1 and v 2 mathbb r n rightarrow mathbb r n 2 are contractions matrix convex sets are sets which are closed under matrix convex combinations a key feature of matrix convex combinations is that the g tuples x y and v 1 xv 1 v 2 y v 2 do not need to have the same size as a result matrix convex sets are a dimension free analog of convex sets while in the classical setting there is only one notion of an extreme point there are three main notions of extreme points for matrix convex sets ordinary matrix and absolute extreme points absolute extreme points are closely related to the classical arveson boundary a central goal in the theory of matrix convex sets is to determine if one of these types of extreme points for a matrix convex set minimally recovers the set through matrix convex combinations this article shows that every real compact matrix convex set which is defined by a linear matrix inequality is the matrix convex hull of its absolute extreme points and that the absolute extreme points are the minimal set with this property furthermore we give an algorithm which expresses a tuple as a matrix convex combination of absolute extreme points with optimal bounds similar results hold when working over the field of complex numbers rather than the reals

every convex free basic semi algebraic set has an lmi representation the matricial solution set of a linear matrix inequality lmi is a convex free basic open semi algebraic set the main theorem of this paper is a converse each such set arises from some lmi the result has implications for semi definite programming and systems engineering as well as for free semi algebraic geometry

subspaces of c algebras abstract basic properties of matricially normed spaces are considered and a simple matrix norm characterization of the subspaces of c algebras is given the latter result is used to study the haagerup tensor products and quotients of such subspaces