data record size requirements for adaptive space time ds cdma signal detection we investigate the data record size requirements of sample matrix inversion based minimum variance distortionless response and maximum signal to interference plus noise ratio adaptive algorithms to meet a given performance objective in joint space time signal detection problems for direct sequence code division multiple access systems we derive closed form expressions that provide the data record size that is necessary to achieve a given performance confidence level in a neighborhood of the optimal performance point as well as expressions that identify the performance level that can be reached for a given data record size this is done by using close approximations of the involved probability density functions the practical significance of the derived expressions lies in the fact that the expressions are functions of the number of antenna elements the number of multipaths and the system spreading gain only while they depend neither on the ideal input covariance matrix which is not known in most realistic applications nor on the exact ideal performance value

yang baxter deformations and rack cohomology abstract in his study of quantum groups drinfeld suggested to consider set theoretic solutions of the yang baxter equation as a discrete analogon as a typical example every conjugacy class in a group or more generally every rack q provides such a yang baxter operator cq x y 7 y x y in this article we study deformations of cq within the space of yang baxter operators infinitesimally these deformations are classified by yang baxter cohomology we show that the yang baxter cochain complex of cq homotopyretracts to a much smaller subcomplex called quasi diagonal this greatly simplifies the deformation theory of cq including the modular case which had previously been left in suspense by establishing that every deformation of cq is gauge equivalent to a quasi diagonal one in a quasi diagonal deformation only behaviourally equivalent elements of q interact if all elements of q are behaviourally distinct then the yang baxter cohomology of cq collapses to its diagonal part which we identify with rack cohomology this establishes a strong relationship between the classical deformation theory following gerstenhaber and the more recent cohomology theory of racks both of which have numerous applications in knot theory

traces in monoidal categories the main result of this paper is the construction of a trace and a trace pairing for endomorphisms satisfying suitable conditions in a monoidal category this construction is a common generalization of the trace for endomorphisms of dualizable objects in a balanced monoidal category and the trace of nuclear operators on a locally convex topological vector space with the approximation property

categorical formulation of finite dimensional quantum algebras we describe how frobenius monoids give the correct categorical description of certain kinds of finite dimensional quantum algebras we develop the concept of an involution monoid and use it to construct a correspondence between finite dimensional c algebras and certain types of frobenius monoids in the category of hilbert spaces using this technology we recast the spectral theorems for commutative c algebras and for normal operators into an explicitly categorical language and we examine the case that the results of measurements do not form finite sets but rather objects in a finite boolean topos we describe the relevance of these results for topological quantum field theory

the geometry of tensor calculus i this paper defines and proves the correctness of the appropriate string diagrams for various kinds of monoidal categories with duals mathematics subject classifications 1991 18d10 52b11 53a45 57m25 68q10 82b23

pareto optimization scheduling with two competing agents to minimize the number of tardy jobs and the maximum cost two agent pareto optimization scheduling the number of tardy jobs and the maximum cost optimal and pareto optimal schedules strongly polynomial time algorithm this paper investigates the pareto optimization scheduling problem on a single machine with two competing agents a and b in which agent a wants to minimize the number of tardy a jobs and agent b wants to minimize the maximum cost of b jobs in the literature the constrained optimization problem of minimizing the number of tardy a jobs under the restriction that the maximum cost of b jobs is bounded is solved in polynomial time this implies that the corresponding pareto optimization scheduling problem can be solved in a weakly polynomial time in this paper by presenting a new algorithm for the constrained optimization problem we provide a strongly polynomial time algorithm for the corresponding pareto optimization scheduling problem experimentation results show that the proposed algorithm for the considered problem is efficient

batalin vilkovisky algebras and two dimensional topological field theories by a batalin vilkovisky algebra we mean a graded commutative algebraa together with an operator a a 1 such that 2 0 and a a is a graded derivation ofa for alla a in this article we show that there is a natural structure of a batalin vilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions we make use of a technique from algebraic topology the theory of operads

multi agent scheduling on a single machine with max form criteria we consider multi agent scheduling on a single machine where the objective functions of the agents are of the max form for the feasibility model we show that the problem can be solved in polynomial time even when the jobs are subject to precedence restrictions for the minimality model we show that the problem is strongly np hard in general but can be solved in pseudo polynomial time when the number of agents is a given constant we then identify some special cases of the minimality model that can be solved in polynomial time

a finite tame wild trichotomy theorem for tensor diagrams abstract in this paper we consider the problem of determining when two tensor networks are equivalent under a heterogeneous change of basis in particular to a tensor or string diagram in a certain monoidal category which we call tensor diagrams we formulate an associated abelian category of representations each representation corresponds to a tensor network on that diagram we then classify which tensor diagrams give rise to categories that are finite tame or wild in the traditional sense of representation theory for those tensor diagrams of finite and tame type we classify the indecomposable representations our main result is that a tensor diagram is wild if and only if it contains a vertex of degree at least three otherwise it is of tame or finite type

non cyclotomic fusion categories etingof nikshych and ostrik ask in 7 2 if every fusion category can be completely defined over a cyclotomic field we show that this is not the case in particular one of the fusion categories coming from the haagerup subfactor 1 and one coming from the newly constructed extended haagerup subfactor 2 can not be completely defined over a cyclotomic field on the other hand we show that the drinfel d center of the even part of the haagerup subfactor is completely defined over a cyclotomic field we identify the minimal field of definition for each of these fusion categories compute the galois groups and identify their galois conjugates ams classification 18d10 46l37

categorical quantum circuits in this paper we extend previous work carried out on the application of the mathematics of category theory to quantum information science specifically we present a realization of a dagger compact category that can model finite dimensional quantum systems and explicitly allows for the interaction of systems of arbitrary possibly unequal dimensions hence our framework can handle generic tensor network states including matrix product states our categorical model subsumes the traditional quantum circuit model while remaining directly and easily applicable to problems stated in the language of quantum information science the circuit diagrams themselves now become morphisms in a category making quantum circuits a special case of a much more general mathematical framework we introduce the key algebraic properties of our tensor calculus diagrammatically and show how they can be applied to solve problems in the field of quantum information

categories of representations of coalgebroids abstract fundamental results of tannaka duality include the reconstruction of a coalgebra in the category of vector spaces from its category of representations equipped with the forgetful functor and the characterization of those categories equipped with a functor into the category of vector spaces which are equivalent to the category of representations of some coalgebra this paper generalizes these results by replacing the category of vector spaces by an arbitrary monoidal category and replacing coalgebras by their several object analogue applications include providing sufficient conditions on a small r linear category to ensure that its category of presheaves is equivalent to the category of representations of some r coalgebra in particular we construct an r coalgebra whose category of representations is equivalent to the category of chain complexes over r

a synthetic approach to markov kernels conditional independence and theorems on sufficient statistics abstract we develop markov categories as a framework for synthetic probability and statistics following work of golubtsov as well as cho and jacobs this means that we treat the following concepts in purely abstract categorical terms conditioning and disintegration various versions of conditional independence and its standard properties conditional products almost surely sufficient statistics versions of theorems on sufficient statistics due to fisher neyman basu and bahadur besides the conceptual clarity offered by our categorical setup its main advantage is that it provides a uniform treatment of various types of probability theory including discrete probability theory measure theoretic probability with general measurable spaces gaussian probability stochastic processes of either of these kinds and many others

pictures of complete positivity in arbitrary dimension two fundamental contributions to categorical quantum mechanics are presented first we generalize the cpm construction that turns any dagger compact category into one with completely positive maps to arbitrary dimension second we axiomatize when a given category is the result of this construction

computing the tutte polynomial of lattice path matroids using determinantal circuits we give a quantum inspired o n 4 algorithm computing the tutte polynomial of a lattice path matroid where n is the size of the ground set of the matroid furthermore this can be improved to o n 2 arithmetic operations if we evaluate the tutte polynomial on a given input fixing the values of the variables the best existing algorithm found in 2004 was o n 5 and the problem has only been known to be polynomial time since 2003 conceptually our algorithm embeds the computation in a determinant using a recently demonstrated equivalence of categories useful for counting problems such as those that appear in simulating quantum systems

algebraic aspects of the quantum yang baxter equation in this paper we examine a variety of algebraic contexts in which the quantum yang baxter equation arises and derive methods for generating new solutions from given ones the solutions we describe are encoded in objects which have a module and a comodule structure over a bialgebra our work here is based in part on the ideas of dr1 dr2 suppose that r m m m m is a solution to the quantum yang baxter equation where m is a finite dimensional vector space over a field k in dr2 it was shown that r can be derived from a left h module structure and a right h comodule structure on m for some bialgebra h over k also see yet maj1 maj2 frt2 the module and comodule structure satisfy a natural compatibility condition m together with these structures is called a left quantum yang baxter h module every left quantum yang baxter module gives rise to a solution to the quantum yang baxter equation in a natural way we study the category hyb of left quantum yang baxter h modules and show that many of the familiar constructions in the module category hm can be made in hyb through these constructions solutions to the quantum yang baxter equation can be combined in various ways to generate new ones we place particular emphasis on the tensor product of objects of hyb and ways in which the tensor algebra t m of an object m of hyb can be viewed as an object of hyb using the tensor algebra we are able to extend certain solutions when dim m 2 to the quantum plane we turn the tensor algebra t m of an object m of hyb into an object of hyb by giving it a module algebra and a comodule algebra structure module algebra and comodule algebra structures play a very important role in this paper in section 9 we develop a calculus for uq sl2 starting with a right k slq 2 comodule algebra structure on the quantum plane k x y q where k slq 2 is the coordinate ring of quantum slq 2 over k this calculus has been described in other ways m s sud the right comodule algebra structure on k x y q accounts for the left uq sl2 module algebra structure on k x y q described in m s we show how the basic irreducible representations of tak follow by the calculus a module algebra action or comodule algebra action is a natural vehicle for extending structure on a submodule or subcomodule respectively to the subalgebra it generates the calculus and the solutions to the quantum yang baxter equation we describe in this paper are determined in this manner on small structures two dimensional in this case we believe that many more results will follow from this circle of ideas in section 2 we discuss basic aspects of coalgebra and bialgebras used in the sequel a good general reference for this is sweedler s book sb on hopf algebras in section 3 we review the quantum yang baxter equation in connection with quantum yang baxter modules the a r construction of faddeev reshetikhin and takhtajan frt is considered in this light and examined in detail in when r arises from two commuting operators even though this particular r is rather simple the algebra structure of a r is surprisingly complex using the scratchpad computer algebra system we determine the algebra structure of a r in an interesting specific case showing that our particular example has a finite non commutative groeb

on modules associated to coalgebra galois extensions abstract for a given entwining structure a c involving an algebraa a coalgebrac and an entwining map c a a c a categorymca of right a c modules is defined and its structure analysed in particular the notion of a measuring of a c to a c is introduced and certain functors betweenmca andmca induced by such a measuring are defined it is shown that these functors are inverse equivalences iff they are exact or one of them faithfully exact and the measuring satisfies a certain galois type condition next left moduleseand right modules associated to ac galois extensionaofbare defined these can be thought of as objects dual to fibre bundles with coalgebracin the place of a structure group and a fibrev cross sections of such associated modules are defined as module mapse bor b it is shown that they can be identified with suitably equivariant maps from the fibre toa also it is shown that ac galois extension is cleft if and only ifa b cas leftb modules and rightc comodules the relationship between the moduleseand is studied in the case whenvis finite dimensional and in the case when the canonical entwining map is bijective

adjunctions and defects in landau ginzburg models abstract we study the bicategory of landau ginzburg models which has polynomials as objects and matrix factorisations as 1 morphisms our main result is the existence of adjoints in this bicategory and formulas for the evaluation and coevaluation maps in terms of atiyah classes and homological perturbation the bicategorical perspective offers a unified approach to landau ginzburg models we show how to compute arbitrary correlators and recover the full structure of open closed tft including the kapustin li disc correlator and a simple proof of the cardy condition in terms of defect operators which in turn are directly computable from the adjunctions

the hom yang baxter equation and hom lie algebras motivated by recent work on hom lie algebras a twisted version of the yang baxter equation called the hom yang baxter equation hybe was introduced by yau j phys a 42 165202 2009 in this paper several more classes of solutions of the hybe are constructed some of the solutions of the hybe are closely related to the quantum enveloping algebra of sl 2 the jones conway polynomial and yetter drinfel d modules under some invertibility conditions we construct a new infinite sequence of solutions of the hybe from a given one

multi agent scheduling on a single machine to minimize total weighted number of tardy jobs we consider the feasibility model of multi agent scheduling on a single machine where each agent s objective function is to minimize the total weighted number of tardy jobs we show that the problem is strongly np complete in general when the number of agents is fixed we first show that the problem can be solved in pseudo polynomial time for integral weights and can be solved in polynomial time for unit weights then we present a fully polynomial time approximation scheme for the problem

performance analysis of beamformers using generalized loading of the covariance matrix in the presence of random steering vector errors robust adaptive beamforming is a key issue in array applications where there exist uncertainties about the steering vector of interest diagonal loading is one of the most popular techniques to improve robustness in this paper we present a theoretical analysis of the signal to interference plus noise ratio sinr for the class of beamformers based on generalized i e not necessarily diagonal loading of the covariance matrix in the presence of random steering vector errors a closed form expression for the sinr is derived that is shown to accurately predict the sinr obtained in simulations this theoretical formula is valid for any loading matrix it provides insights into the influence of the loading matrix and can serve as a helpful guide to select it finally the analysis enables us to predict the level of uncertainties up to which robust beamformers are effective and then depart from the optimal sinr

a 2 categorical pasting theorem in order to facilitate the study of 2 categories with structure we state and prove an n categorical pasting theorem this is based upon a new definition of n pasting scheme that generalises johnson s definition of a well formed loop free pasting scheme by weakening his no direct loops condition we define n pasting prove the theorem and show that for n 3 it incorporates all possible composites of n cells we show that that is not true for higher n we define the horizontal n category of an n 1 category to generalise that of a 2 category we define horizontal and vertical composition for an n 1 category and we state and prove an interchange law we also study further conditions on a pasting diagram and their impact upon how one may evaluate a composite and we express street s free n categories in terms of left adjoints

a compositional approach to quantum functions we introduce a notion of quantum function and develop a compositional framework for finite quantum set theory based on a 2 category of quantum sets and quantum functions we use this framework to formulate a 2 categorical theory of quantum graphs which captures the quantum graphs and quantum graph homomorphisms recently discovered in the study of nonlocal games and zero error communication and relates them to quantum automorphism groups of graphs considered in the setting of compact quantum groups we show that the 2 categories of quantum sets and quantum graphs are semisimple we analyze dualisable and invertible 1 morphisms in these 2 categories and show that they correspond precisely to the existing notions of quantum isomorphism and classical isomorphism between sets and graphs

higher dimensional algebra iv 2 tangles just as knots and links can be algebraically described as certain morphisms in the category of tangles in 3 dimensions compact surfaces smoothly embedded in r4 can be described as certain 2 morphisms in the 2 category of 2 tangles in 4 dimensions using the work of carter rieger and saito we prove that this 2 category is the free semistrict braided monoidal 2 category with duals on one unframed self dual object by this universal property any unframed self dual object in a braided monoidal 2 category with duals determines an invariant of 2 tangles in 4 dimensions

double quantum groups the construction of the drinfeld double d h of a finite dimensional hopf algebra h was one of the first examples of a quasitriangular hopf algebra whose category of modules md h is braided the braided category of yetter drinfeld modules ydhh is the analogue for infinite dimensional hopf algebras it uses a strong dependence between the h module and the h comodule structures we generalize this construction to the category mca of entwined modules that is a modules and c comodules over hopf algebras a and c where the structures are only related by an entwining map c a a c we show that mca is braided iff there is an r map r c c a a satisfying suitable axioms that generalize the axioms of an r matrix for finite dimensional c there is a quasitriangular hopf algebra structure on hom c a the quantum group double generalizing the construction of the drinfeld double

generalized counting constraint satisfaction problems with determinantal circuits abstract generalized counting constraint satisfaction problems include holant problems with planarity restrictions polynomial time algorithms for such problems include matchgates and matchcircuits which are based on pfaffians in particular they use gates which are expressible in terms of a vector of sub pfaffians of a skew symmetric matrix we introduce a new type of circuit based instead on determinants with seemingly different expressive power in these determinantal circuits a gate is represented by the vector of all minors of an arbitrary matrix determinantal circuits permit a different class of gates applications of these circuits include proofs of theorems from algebraic graph theory including the chung langlands formula for the number of rooted spanning forests of a graph and computing tutte polynomials of certain matroids they also give a strategy for simulating quantum circuits with closed timelike curves monoidal category theory provides a useful language for discussing such counting problems turning combinatorial restrictions into categorical properties we introduce the counting problem in monoidal categories and count preserving functors as a way to study fp subclasses of problems in settings which are generally p hard using this machinery we show that surprisingly determinantal circuits can be simulated by pfaffian circuits at quadratic cost

frobenius monads and pseudomonoids six equivalent definitions of frobenius algebra in a monoidal category are provided in a monoidal bicategory a pseudoalgebra is frobenius if and only if it is star autonomous autonomous pseudoalgebras are also frobenius what it means for a morphism of a bicategory to be a projective equivalence is defined this concept is related to strongly separable frobenius algebras and weak monoidal morita equivalence wreath products of frobenius algebras are discussed

recursive short data record estimation of av and mmse mvdr linear filters for ds cdma antenna array systems the presence of the desired signal during estimation of the minimum mean square error mmse minimum variance distortionless response mvdr and auxiliary vector av filters under limited data support leads to significant signal to interference plus noise ratio sinr performance degradation we quantify this observation in the context of direct sequence code division multiple access ds cdma communications by deriving close approximations for the mean square filter estimation error the probability density function of the output sinr and the probability density function of the symbol error rate ser of the sample matrix inversion smi receiver evaluated using both a desired signal present and desired signal absent input covariance matrix to avoid such performance degradation we propose a ds cdma receiver that utilizes a simple pilot assisted algorithm that estimates and then subtracts the desired signal component from the received signal prior to filter estimation then to accommodate decision directed operation we develop two recursive algorithms for the on line estimation of the av and mmse mvdr filter and we study their convergence properties finally simulation studies illustrate the ser performance of the overall receiver structures

modular operads and the nerve theorem abstract we describe a category of undirected graphs which comes equipped with a faithful functor into the category of colored modular operads the associated singular functor from modular operads to presheaves is fully faithful and its essential image can be classified by a segal condition this theorem can be used to recover a related statement due to andre joyal and joachim kock concerning a larger category of undirected graphs whose functor to modular operads is not just faithful but also full

higher algebraic structures and quantization we derive quasi quantum groups in 2 1 dimensional topological field theory directly from the classical action and the path integral detailed computations are carried out for the chern simons theory with finite gauge group the principles behind our computations are presumably more general we extend the classical action in ad 1 dimensional topological theory to manifolds of dimension less thand 1 we then construct a generalized path integral which ind 1 dimensions reduces to the standard one and ind dimensions reproduces the quantum hilbert space in a 2 1 dimensional topological theory the path integral over the circle is the category of representations of a quasi quantum group in this paper we only consider finite theories in which the generalized path integral reduces to a finite sum new ideas are needed to extend beyond the finite theories treated here

an alternate derivation of the distribution of the conditioned signal to noise ratio a result in multiple regression analysis is used to derive the probability distribution of the conditioned signal to noise ratio

quantum koszul complexes for majid s braided lie algebras a yang baxter operator was canonically associated to any cocommutative braided lie algebra as defined by majid applying previous results we construct quantum koszul complexes for braided lie algebras with respect to this operator in the usual case the complexes reduce to the chevalley eilenberg complexes for classical lie algebras

two dimensional algebra in lattice gauge theory we provide a visual and intuitive introduction to effectively calculating in 2 groups along with explicit examples coming from non abelian 1 and 2 form gauge theory in particular we utilize string diagrams tools similar to tensor networks to compute the parallel transport along a surface using approximations on a lattice we prove a convergence theorem for the surface transport in the continuum limit locality is used to define infinitesimal parallel transport and two dimensional algebra is used to derive finite versions along arbitrary surfaces with sufficient orientation data the correct surface ordering is dictated by two dimensional algebra and leads to an interesting diagrammatic picture for gauge fields interacting with particles and strings on a lattice the surface ordering is inherently complicated but we prove a simplification theorem confirming earlier results of schreiber and waldorf assuming little background we present a simple way to understand some abstract concepts of higher category theory in doing so we review all the necessary categorical concepts from the tensor network point of view as well as many aspects of higher gauge theory we provide a visual and intuitive introduction to effectively calculating in 2 groups along with explicit examples coming from non abelian 1 and 2 form gauge theory in particular we utilize string diagrams tools similar to tensor networks to compute the parallel transport along a surface using approximations on a lattice we prove a convergence theorem for the surface transport in the continuum limit locality is used to define infinitesimal parallel transport and two dimensional algebra is used to derive finite versions along arbitrary surfaces with sufficient orientation data the correct surface ordering is dictated by two dimensional algebra and leads to an interesting diagrammatic picture for gauge fields interacting with particles and strings on a lattice the surface ordering is inherently complicated but we prove a simplification theorem confirming earlier results of schreiber and waldorf assuming little background we present a simple way to understand some abstract concepts of higher cat

actions of monoidal categories and generalized hopf smash products let r be a k algebra and c a monoidal category assume given the structure of a c category on the category rm of left r modules that is the monoidal category c is assumed to act on the category rm by a coherently associative bifunctor c rm rm we assume that this bifunctor is right exact in its right argument in this setup we show that every algebra a respectively coalgebra c in c gives rise to an r ring a r respectively an r coring c r whose modules respectively comodules are the a modules respectively c comodules within the category rm we show that this very general scheme for constructing co associative co rings gives conceptual explanations for the double of a quasi hopf algebra as well as certain doubles of hopf algebras in braided categories each time avoiding ad hoc computations showing associativity

correspondences of ribbon categories much of algebra and representation theory can be formulated in the general framework of tensor categories the aim of this paper is to further develop this theory for braided tensor categories several results are established that do not have a substantial counterpart for symmetric tensor categories in particular we exhibit various equivalences involving categories of modules over algebras in ribbon categories finally we establish a correspondence of ribbon categories that can be applied to and is in fact motivated by the coset construction in conformal quantum field theory

linear logic coherence and dinaturality a general coherence theorem for monoidal closed structures is obtained by modifying the logical approach to coherence questions due to lambek 1969 1990 by making use of linear logic linear logic introduced by girard has many advantages which are of use in studying coherence most notably its resource sensitive nature makes it ideal for studying monoidal closed structures the logical approach is also modified by using natural deduction rather than sequent calculus the natural deduction system in question is proof nets also introduced by girard proof nets have several important properties which are exploited to prove the coherence theorem in particular the cut elimination procedure is confluent and strongly normalizing the approach to coherence is to define a general structure the autonomous deductive system for defining many theories of monoidal closed categories an autonomous deductive system is a deductive system with several added features which are suggested by the properties of proof nets it is then possible to give a straightforward criterion for whether a given theory of monoidal closed categories specified by an autonomous deductive system is coherent finally a relationship is established between coherence and the composition problem for dinatural transformations thus the dinatural approach to modelling polymorphic types due to bainbridge et al 1990 can be extended to linear polymorphism

design and analysis of supervised and decision directed estimators of the mmse lcmv filter in data limited environments we consider sample matrix inversion smi type estimates of the minimum mean square error mmse and the linearly constrained minimum variance lcmv linear filters obtained from data records of limited size we quantify theoretically the detrimental effect of the desired signal energy level on the mean square ms filter estimation error and the normalized output signal to interference plus noise ratio sinr by deriving a new exact analytical expression and a lower bound respectively for cases where accumulation of pure disturbance observations is not possible we show theoretically how certain intuitive pilot assisted and decision directed adaptive filter implementations that utilize desired signal present data observations perform close to their desired signal absent counterparts simulation studies illustrate our theoretical developments in the context of spread spectrum communications over multipath fading channels under perfect and nonperfect synchronization

a graphical category for higher modular operads abstract we present a homotopy theory for a weak version of modular operads whose compositions and contractions are only defined up to homotopy this homotopy theory takes the form of a quillen model structure on the collection of simplicial presheaves for a certain category of undirected graphs this new category of undirected graphs denoted u plays a similar role for modular operads that the dendroidal category plays for operads we carefully study properties of u including the existence of certain factorization systems related structures such as cyclic operads and stable modular operads can be similarly treated using categories derived from u

on the rosenberg zelinsky sequence in abelian monoidal categories we consider frobenius algebras and their bimodules in certain abelian monoidal categories in particular we study the picard group of the category of bimodules over a frobenius algebra i e the group of isomorphism classes of invertible bimodules the rosenberg zelinsky sequence describes a homomorphism from the group of algebra automorphisms to the picard group which however is typically not surjective we investigate under which conditions there exists a morita equivalent frobenius algebra for which the corresponding homomorphism is surjective one motivation for our considerations is the orbifold construction in conformal field theory

on the geometry and algebra of networks with state in 1 an algebra of automata with interfaces span graph was introduced with main operation being communicating parallel composition a system is represented by an expression in this algebra a system so described has two aspects an informal network geometry arising from the algebraic expression and a space of states and transition given by its evaluation in span graph note that span graph yields purely compositional descriptions of systems in this paper we give an alternative globally non compositional view of the same systems in order to do this we make mathematically precise the network geometry in terms of monoidal graphs and assignment of state in terms of morphisms of monoidal graphs we call such globally described systems networks with state to relate networks with state and span graph systems we use the algebra of cospans of monoidal graphs as an illustration we give a new representation of a non compositionally described class of petri nets in the compositional setting of span graph we also present a tool for calculating with networks with state both algebras of spans and of cospans are symmetric monoidal categories with commutative separable algebra structures on the objects we include a short section following 2 in which we show that a simple modification of the algebra allows the description of networks in which the tangling of connectors may be represented yielding a connection with the theory of knots