limits for entanglement measures the basic principle of entanglement processing says that entanglement cannot increase under local operations and classical communication based on this principle we show that any entanglement measure e suitable for the regime of a high number of identically prepared entangled pairs satisfies ed or e or ef where ed and ef are the entanglement of distillation and formation respectively moreover we exhibit a theorem establishing a very general form of bounds for distillable entanglement

product structure of the fuzzy n ary factor polygroup fuzzy n ary hypergroups were introduced as suitable generalizations of fuzzy polygroups and a special case of fuzzy n ary hyper groups the aim of this paper is to introduce the notion of fuzzy n ary factor polygroups of a polygroup based on the fuzzy n ary factor group we also study the product structures of the generating fuzzy factor n ary groups at the end of the paper we prove the fundamental theorem of isomorphism of fuzzy n ary groups

strongly transitive geometric spaces associated to hypermodules abstract in this paper we use the strongly regular relation on hypermodules with canonical hypergroup over a given krasner hyperring in this way we consider the fundamental relation defined on a hypermodule and prove some results in this respect also we determine a family p m of subsets of a hypermodule m and we give sufficient conditions such that the geometric space m p m is strongly transitive and the relation is transitive

quasi doubly stochastic operator on l1 and nielsen s theorem in this paper we introduce the quasidoubly stochastic operator which is between doubly stochastic operators and column stochastic operators so as to apply to characterized operator s on l1 such that sf is majorized by f for every f l1 we present some classes of majorization preservers on l1 under quasi doubly stochastic operators moreover as an application of our result in quantum physics the convertibility of pure states of a composite system by local operations and classical communication has been considered

quantum data hiding we expand on our work on quantum data hiding hiding classical data among parties who are restricted to performing only local quantum operations and classical communication locc we review our scheme that hides one bit between two parties using bell 1964 states and we derive upper and lower bounds on the secrecy of the hiding scheme we provide an explicit bound showing that multiple bits can be hidden bitwise with our scheme we give a preparation of the hiding states as an efficient quantum computation that uses at most one ebit of entanglement a candidate data hiding scheme that does not use entanglement is presented we show how our scheme for quantum data hiding can be used in a conditionally secure quantum bit commitment scheme

applications of interval valued fuzzy n ary polygroups with respect to t norms t conorms using the triangular norm t conorm s in the context of n ary polygroups we introduce the concept of an interval valued anti fuzzy n ary subpolygroup with respect to t s respectively a necessary and sufficient condition for an interval valued fuzzy subset in order to be an interval valued anti fuzzy n ary subpolygroup is established and some important results are presented this study is useful for the most computational part of fuzzy control which is the defuzzification

asymptotic state discrimination and a strict hierarchy in distinguishability norms in this paper we consider the problem of discriminating quantum states by local operations and classical communication locc when an arbitrarily small amount of error is permitted this paradigm is known as asymptotic state discrimination and we derive necessary conditions for when two multipartite states of any size can be discriminated perfectly by asymptotic locc we use this new criterion to prove a gap in the locc and separable distinguishability norms we then turn to the operational advantage of using two way classical communication over one way communication in locc processing with a simple two qubit product state ensemble we demonstrate a strict majorization of the two way locc norm over the one way norm

a semidefinite program for distillable entanglement we show that the maximum fidelity obtained by a positive partial transpose p p t distillation protocol is given by the solution to a certain semidefinite program this gives a number of new lower and upper bounds on p p t distillable entanglement and thus new upper bounds on 2 locally distillable entanglement in the presence of symmetry the semidefinite program simplifies considerably becoming a linear program in the case of isotropic and werner states using these techniques we determine the p p t distillable entanglement of asymmetric werner states and maximally correlated states we conclude with a discussion of possible applications of semidefinite programming to quantum codes and 1 local distillation

tight asymptotic bounds on local hypothesis testing between a pure bipartite state and the white noise state we consider asymptotic hypothesis testing or state discrimination with asymmetric treatment of errors between an arbitrary fixed bipartite pure state psi rangle and the white noise state the completely mixed state under one way local operations and classical communications locc two way locc and separable positive operator valued measures povms as a result we derive the hoeffding bounds under two way locc povms and separable povms further we derive stein s lemma type of optimal error exponents under one way locc two way locc and separable povms up to the third order which clarifies the difference between one way and two way locc povm our results clarify the relationship between the entanglement of renyi entropy and the hypothesis testing under locc since the entanglement of renyi entropy appears in the formula of both the hoeffding bounds and stein s lemma type of error exponents this paper gives a very rare example in which the optimal performance under the infinite round two way locc is also equal to that under separable operations and can be attained with two round communication but not with the one way locc

classification of mixed three qubit states we introduce a classification of mixed three qubit states in which we define the classes of separable biseparable w and greenberger horne zeilinger states these classes are successively embedded into each other we show that contrary to pure w type states the mixed w class is not of measure zero we construct witness operators that detect the class of a mixed state we discuss the conjecture that all entangled states with positive partial transpose pptes belong to the w class finally we present a new family of pptes edge states with maximal ranks

ultimate data hiding in quantum mechanics and beyond the phenomenon of data hiding i e the existence of pairs of states of a bipartite system that are perfectly distinguishable via general entangled measurements yet almost indistinguishable under locc is a distinctive signature of nonclassicality the relevant figure of merit is the maximal ratio called data hiding ratio between the distinguishability norms associated with the two sets of measurements we are comparing typically all measurements vs locc protocols for a bipartite n times n quantum system it is known that the data hiding ratio scales as n i e the square root of the real dimension of the local state space of density matrices we show that for bipartite n a times n b systems the maximum data hiding ratio against locc protocols is theta left min n a n b right this scaling is better than the previously obtained upper bounds o left sqrt n a n b right and min n a 2 n b 2 and moreover our intuitive argument yields constants close to optimal in this paper we investigate data hiding in the more general context of general probabilistic theories gpts an axiomatic framework for physical theories encompassing only the most basic requirements about the predictive power of the theory the main result of the paper is the determination of the maximal data hiding ratio obtainable in an arbitrary gpt which is shown to scale linearly in the minimum of the local dimensions we exhibit an explicit model achieving this bound up to additive constants finding that quantum mechanical data hiding ratio is only of the order of the square root of the maximal one our proof rests crucially on an unexpected link between data hiding and the theory of projective and injective tensor products of banach spaces finally we develop a body of techniques to compute data hiding ratios for a variety of restricted classes of gpts that support further symmetries

causal limit on quantum communication the capacity of a channel is known to be equivalent to the highest rate at which it can generate entanglement analogous to entanglement the notion of a causality measure characterizes the temporal aspect of quantum correlations despite holding an equally fundamental role in physics temporal quantum correlations have yet to find their operational significance in quantum communication here we uncover a connection between quantum causality and channel capacity we show the amount of temporal correlations between two ends of the noisy quantum channel as quantified by a causality measure implies a general upper bound on its channel capacity the expression of this new bound is simpler to evaluate than most previously known bounds we demonstrate the utility of this bound by applying it to a class of shifted depolarizing channels which results in improvement over previously known bounds for this class of channels

on the chow motive of some moduli spaces we study the motive of moduli spaces of stable vector bundles over a smooth projective curve we prove this motive lies in the category generated by the motive of the curve and we compute its class in the grothendieck ring of the category of motives as applications we compute the poincare hodge polynomials and the number of points over a finite field and we study some conjectures on algebraic cycles on these moduli spaces

self catalytic conversion of pure quantum states conversion of entangled states under stochastic local operations and classical communication s locc admits the phenomenon of catalysis here we explore the possibility of a copy of the initial state itself performing as a catalyst which we call a self catalytic process we show explicit examples of self catalysis necessary and sufficient conditions for the phenomenon to take place are discussed we numerically estimate how frequent it is and we show that increasing the number of copies used as catalyst can increase the probability of conversion but does not make the process deterministic by the end we conjecture that under locc the probability of finding a self catalytic reaction does not increase monotonically with the dimensions whereas under slocc it does increase

condition on the renyi entanglement entropy under stochastic local manipulation the r enyi entanglement entropy ree is an entanglement quantifier considered as a natural generalization of the entanglement entropy when it comes to stochastic local operations and classical communication slocc however only a limited class of the rees satisfy the monotonicity condition while their statistical properties beyond mean values have not been fully investigated here we establish a general condition that the probability distribution of the ree of any order obeys under slocc the condition is obtained by introducing a family of entanglement monotones that contain the higher order moments of the rees the contribution from the higher order moments imposes a strict limitation on entanglement distillation via slocc we find that the upper bound on success probabilities for entanglement distillation exponentially decreases as the amount of raised entanglement increases which cannot be captured from the monotonicity of the ree based on the strong restriction on entanglement transformation under slocc we design a new method to estimate entanglement in quantum many body systems from experimentally observable quantities

bounds on entanglement distillation and secret key agreement for quantum broadcast channels the squashed entanglement of a quantum channel is an additive function of quantum channels which finds application as an upper bound on the rate at which secret key and entanglement can be generated when using a quantum channel a large number of times in addition to unlimited classical communication this quantity has led to an upper bound of log 1 eta 1 eta on the capacity of a pure loss bosonic channel for such a task where eta is the average fraction of photons that make it from the input to the output of the channel the purpose of this paper is to extend these results beyond the single sender single receiver setting to the more general case of a single sender and multiple receivers a quantum broadcast channel we employ multipartite generalizations of the squashed entanglement to constrain the rates at which secret key and entanglement can be generated between any subset of the users of such a channel along the way developing several new properties of these measures we apply our results to the case of a pure loss broadcast channel with one sender and two receivers

additivity of the classical capacity of entanglement breaking quantum channels we show that for the tensor product of an entanglement breaking quantum channel with an arbitrary quantum channel both the minimum entropy of an output of the channel and the holevo schumacher westmoreland capacity are additive in addition for the tensor product of two arbitrary quantum channels we give a bound involving entanglement of formation for the amount of subadditivity for minimum entropy output or superadditivity for classical capacity that can occur

entangled pure state transformations via local operations assisted by finitely many rounds of classical communication we consider generic pure n qubit states and a general class of pure states of arbitrary dimensions and arbitrarily many subsystems we characterize those states which can be reached from some other state via local operations assisted by finitely many rounds of classical communication locc n for n qubits with n 3 we show that this set of states is of measure zero which implies that the maximally entangled set is generically of full measure if restricted to the practical scenario of locc n moreover we identify a class of states for which any locc n protocol can be realized via a concatenation of deterministic steps we show however that in general there exist state transformations which require a probabilistic step within the protocol which highlights the difference between bipartite and multipartite locc

fully simple semihypergroups transitive digraphs and sequence a000712 abstract fully simple semihypergroups have been introduced in 9 motivated by the study of the transitivity of the fundamental relation in semihypergroups here we determine a transversal of isomorphism classes of fully simple semihypergroups with a right absorbing element the structure of that transversal can be described by means of certain transitive acyclic digraphs moreover we prove that if n is an integer 2 then the number of isomorphism classes of fully simple semihypergroups of size n 1 with a right absorbing element is the n 1 th term of sequence a000712 in 20 namely k 0 n p k p n k where p k denotes the number of nonincreasing partitions of integer k

applications of the α relation to krasner hyperrings abstract this paper deals with a special type of hyperrings called krasner hyperring the main tool is the studding of the relation the relation was introduced by davvaz and vougiouklis indeed we use the relation in a given krasner hyperring in this way the relation is the smallest equivalence relation on a krasner hyperring r such that r is a commutative ring results are obtained on krasner hyperrings derived hyperrings and parts

nilpotent groups derived from hypergroups in this paper we introduce the smallest equivalence relation on a hypergroup h such that the quotient h the set of all equivalence classes is a nilpotent group the characterization of nilpotent groups via strongly regular relations is investigated and several results on the topic are presented

construction of n ary f polygroups by an n ary f hyperoperation on a non empty set a we mean a function from a a to the set of all non zero fuzzy subsets of a where a appears n times using this concept we introduce the notion of n ary f polygroups which is a generalization of ideas presented by zahedi hasankhani and davvaz et al some basic results are obtained

state convertibility in the von neumann algebra framework we establish a generalisation of the fundamental state convertibility theorem in quantum information to the context of bipartite quantum systems modelled by commuting semi finite von neumann algebras namely we establish a generalisation to this setting of nielsen s theorem on the convertibility of quantum states under local operations and classical communication locc schemes along the way we introduce an appropriate generalisation of locc operations and connect the resulting notion of approximate convertibility to the theory of singular numbers and majorisation in von neumann algebras as an application of our result in the setting of mathrm ii 1 factors we show that the entropy of the singular value distribution relative to the unique tracial state is an entanglement monotone in the sense of vidal thus yielding a new way to quantify entanglement in that context building on previous work in the infinite dimensional setting we show that trace vectors play the role of maximally entangled states for general mathrm ii 1 factors examples are drawn from infinite spin chains quasi free representations of the car and discretised versions of the ccr

transposition hypergroups noncommutative join spaces the study of various kinds of algebraic hypergroups is unified in the theory of transposition hypergroups the hyperoperation in any hypergroup has two inverses given bya b x a xb andb a x a bx a transposition hypergroup is a hypergroup where transposition b a c d impliesad bc holds the algebra of transposition hypergroups is developed closed subhypergroupsnthat are reflexive a n n a have distinguished structural significance the quotient space of a transposition hypergroup modulo a reflexive closed subhypergroup forms a transposition hypergroup that is a polygroup then generalizations of the isomorphism theorems and the jordan holder theorem of group theory are obtained

hyperrings and α relations a general approach abstract the study of the equivalence relations of particular multialgebras for which the factor multialgebras are universal algebras satisfying certain identities is a very important and intensively studied topic in multialgebra theory and not only our paper provides a general multialgebraic approach for the construction of all these relations and also some important and interesting properties concerning the construction of the corresponding factor universal algebra one of the purposes of our approach is to improve some results concerning relations of hyperrings and krasner hyperrings and we will do this in the last sections of this paper

strengthened monotonicity of relative entropy via pinched petz recovery map the quantum relative entropy between two states satisfies a monotonicity property meaning that applying the same quantum channel to both states can never increase their relative entropy it is known that this inequality is only tight when there is a recovery map that exactly reverses the effects of the quantum channel on both states in this paper we strengthen this inequality by showing that the difference of relative entropies is bounded below by the measured relative entropy between the first state and a recovered state from its processed version the recovery map is a convex combination of rotated petz recovery maps and perfectly reverses the quantum channel on the second state as a special case we reproduce recent lower bounds on the conditional mutual information such as the one proved by fawzi and renner our proof only relies on the elementary properties of pinching maps and the operator logarithm

products of fuzzy finite state machines abstract we introduce the concepts of fuzzy transformation semigroups coverings cascade and wreath products for fuzzy finite state machines in order to overcome some of the difficulties which arise from the fuzzification of these concepts we also introduce the notions of polysemigroups and weak coverings

transformations of w type entangled states the transformations of w type entangled states by using local operations assisted with classical communication are investigated for this purpose a parametrization of the w type states which remains invariant under local unitary transformations is proposed and a complete characterization of the local operations carried out by a single party is given these are used for deriving the necessary and sufficient conditions for deterministic transformations a convenient upper bound for the maximum probability of distillation of arbitrary target states is also found

fuzzy n ary subpolygroups recently n ary hypergroups were introduced and analyzed by davvaz and vougiouklis n ary polygroups are suitable generalizations of polygroups and a special case of n ary hypergroups the aim of this paper is to introduce the notion of a fuzzy n ary subpolygroup of an n ary polygroup and to extend the fuzzy results of fundamental equivalence relations to n ary polygroups

multivariate trace inequalities we prove several trace inequalities that extend the golden thompson and the araki lieb thirring inequality to arbitrarily many matrices in particular we strengthen lieb s triple matrix inequality as an example application of our four matrix extension of the golden thompson inequality we prove remainder terms for the monotonicity of the quantum relative entropy and strong sub additivity of the von neumann entropy in terms of recoverability we find the first explicit remainder terms that are tight in the commutative case our proofs rely on complex interpolation theory as well as asymptotic spectral pinching providing a transparent approach to treat generic multivariate trace inequalities

tensor powers of 2 positive maps by studying tensor powers n of a positive map of b k into b h for finite dimensional hilbert spaces k and h we give conditions for a map such that n 2 positive for all n to be nondistillable and completely positive

distinguishability of quantum states by separable operations in this paper we study the distinguishability of multipartite quantum states by separable operations we first present a necessary and sufficient condition for a finite set of orthogonal quantum states to be distinguishable by separable operations an analytical version of this condition is derived for the case of d 1 pure states where d is the total dimension of the state space under consideration a number of interesting consequences of this result are then carefully investigated remarkably we show there exists a large class of 2 otimes 2 separable operations not being realizable by local operations and classical communication before our work only a class of 3 otimes 3 nonlocal separable operations was known bennett phys rev a 59 1070 1999 we also show that any basis of the orthogonal complement of a multipartite pure state is indistinguishable by separable operations if and only if this state cannot be a superposition of one or two orthogonal product states i e has an orthogonal schmidt number not less than three thus generalize the recent work about indistinguishable bipartite subspaces watrous phys rev lett 95 080505 2005 notably we obtain an explicit construction of indistinguishable subspaces of dimension 7 or 6 by considering a composite quantum system consisting of two qutrits resp three qubits which is slightly better than the previously known indistinguishable bipartite subspace with dimension 8

fully simple semihypergroups abstract in this paper we consider the class of semihypergroups h such that all subsemihypergroups k h are simple and when k 3 the fundamental relation k is not transitive for these semihypergroups we prove that hyperproducts of elements in h have size 2 and the quotient semigroup h is trivial this last result allows us to completely characterize these semihypergroups in terms of a small set of simple semihypergroups of size 3 finally we solve a problem on strongly simple semihypergroups introduced in 11

pretty strong converse for the quantum capacity of degradable channels we exhibit a possible road toward a strong converse for the quantum capacity of degradable channels in particular we show that all degradable channels obey what we call a pretty strong converse when the code rate increases above the quantum capacity the fidelity makes a discontinuous jump from 1 to at most 1 2 asymptotically a similar result can be shown for the private classical capacity furthermore we can show that if the strong converse holds for symmetric channels which have quantum capacity zero then degradable channels obey the strong converse the above mentioned asymptotic jump of the fidelity at the quantum capacity then decreases from 1 to 0

soft polygroups soft set theory introduced by molodtsov has been considered as an effective mathematical tool for modeling uncertainties in this paper we initiate the study of algebraic hyperstructures of soft sets the concepts of soft polygroups normal soft polygroups soft subpolygroups and normal soft subpolygroups are introduced and some related properties are investigated furthermore we propose the homomorphism and isomorphism of soft polygroups and establish three isomorphism theorems for soft polygroups

positivity of linear maps under tensor powers we investigate linear maps between matrix algebras that remain positive under tensor powers i e under tensoring with n copies of themselves completely positive and completely co positive maps are trivial examples of this kind we show that for every n there exist non trivial maps with this property and that for two dimensional hilbert spaces there is no non trivial map for which this holds for all n for higher dimensions we reduce the existence question of such non trivial tensor stable positive maps to a one parameter family of maps and show that an affirmative answer would imply the existence of non positive partial transpose bound entanglement as an application we show that any tensor stable positive map that is not completely positive yields an upper bound on the quantum channel capacity which for the transposition map gives the well known cb norm bound we furthermore show that the latter is an upper bound even for the local operations and classical communications assisted quantum capacity and that moreover it is a strong converse rate for this task

dynamical entanglement unlike the entanglement of quantum states very little is known about the entanglement of bipartite channels called dynamical entanglement here we work with the partial transpose of a superchannel and use it to define computable measures of dynamical entanglement such as the negativity we show that a version of it the max logarithmic negativity represents the exact asymptotic dynamical entanglement cost we discover a family of dynamical entanglement measures that provide necessary and sufficient conditions for bipartite channel simulation under local operations and classical communication and under operations with positive partial transpose

fuzzy n ary polygroups related to fuzzy points recently fuzzy n ary sub polygroups were introduced and studied by davvaz corsini and leoreanu fotea b davvaz p corsini v leoreanu fotea fuzzy n ary sub polygroups comput math appl 57 2008 141 152 now in this paper the concept of email protected q fuzzy n ary sub polygroups email protected email protected email protected email protected fuzzy n ary sub polygroups and fuzzy n ary sub polygroup with thresholds of an n ary polygroup are introduced and some characterizations are described also we give the definition of implication based fuzzy n ary sub polygroups in an n ary polygroup in particular the implication operators in lukasiewicz system of continuous valued logic are discussed

polygroups derived from cogroups abstract a polygroup is a completely regular reversible in itself multigroup in the sense of dresher and ore 4 these systems occur naturally in the study of algebraic logic 1 2 some algebraic and combinatorial properties were developed in 3 the question of whether or not q 2 group the class of double quotients of groups is the same as the class of all chromatic polygroups is important for many reasons in this paper double quotients of groups are constructed in a natural way from cogroups cogroups were introduced by eaton 5 in an attempt to axiomatize d hypergroups i e systems obtainable from groups by right coset decompositions with respect to not necessarily normal subgroups eaton s axioms apply only to finite systems utumi 7 formulated a general notion and gave an example of a cogroup not isomorphic to a d hypergroup utumi s scalar partition hypergroups appear in section 2 the main results in section 3 show that polygroups derivable from cogroups are chromatic and they include all double quotients of groups

everything you always wanted to know about locc but were afraid to ask in this paper we study the subset of generalized quantum measurements on finite dimensional systems known as local operations and classical communication locc while locc emerges as the natural class of operations in many important quantum information tasks its mathematical structure is complex and difficult to characterize here we provide a precise description of locc and related operational classes in terms of quantum instruments our formalism captures both finite round protocols as well as those that utilize an unbounded number of communication rounds while the set of locc is not topologically closed we show that finite round locc constitutes a compact subset of quantum operations additionally we show the existence of an open ball around the completely depolarizing map that consists entirely of locc implementable maps finally we demonstrate a two qubit map whose action can be approached arbitrarily close using locc but nevertheless cannot be implemented perfectly

minimal order semihypergroups of type u on the right ii abstract we prove a new characterization of the heart w h and of the derived d h of a finite semihypergroup h of type u on the right in terms of its hyperproducts consequently we characterize the minimal size of the proper semihypergroups h of type u on the right such that w h d h h w h d h h or w h d h h moreover we obtain new results on the fundamental relations and on finite semihypergroups of type u on the right in particular we prove that is transitive is transitive if h 8 and that there exist semihypergroups of type u on the right of size 9 where is not transitive hence the minimal size for a semihypergroup of type u on the right to have a nontransitive is 9

the uniqueness theorem for entanglement measures we explore and develop the mathematics of the theory of entanglement measures after a careful review and analysis of definitions of preliminary results and of connections between conditions on entanglement measures we prove a sharpened version of a uniqueness theorem which gives necessary and sufficient conditions for an entanglement measure to coincide with the reduced von neumann entropy on pure states we also prove several versions of a theorem on extreme entanglement measures in the case of mixed states we analyze properties of the asymptotic regularization of entanglement measures proving for example convexity for the entanglement cost and for the regularized relative entropy of entangle ment

unextendible product bases and bound entanglement an unextendible product basis upb for a multipartite quantum system is an incomplete orthogonal product basis whose complementary subspace contains no product state we give examples of upbs and show that the uniform mixed state over the subspace complementary to any upb is a bound entangled state we exhibit a tripartite 2 3 2 3 2 upb whose complementary mixed state has tripartite entanglement but no bipartite entanglement i e all three corresponding 2 3 4 bipartite mixed states are unentangled we show that members of a upb are not perfectly distinguishable by local positive operator valued measurements and classical communication s0031 9007 99 09360 6

semidefinite programming converse bounds for quantum communication we derive several efficiently computable converse bounds for quantum communication over quantum channels in both the one shot and asymptotic regime first we derive one shot semidefinite programming sdp converse bounds on the amount of quantum information that can be transmitted over a single use of a quantum channel which improve the previous bound from tomamichel berta renes nat commun 7 2016 as applications we study quantum communication over depolarizing channels and amplitude damping channels with finite resources second we find an sdp strong converse bound for the quantum capacity of an arbitrary quantum channel which means the fidelity of any sequence of codes with a rate exceeding this bound will vanish exponentially fast as the number of channel uses increases furthermore we prove that the sdp strong converse bound improves the partial transposition bound introduced by holevo and werner third we prove that this sdp strong converse bound is equal to the so called max rains information which is an analog to the rains information introduced in tomamichel wilde winter ieee trans inf theory 63 715 2017 our sdp strong converse bound is weaker than the rains information but it is efficiently computable for general quantum channels

conformal field theory and the cohomology of the moduli space of stable bundles let g be a compact riemann surface of genus g 2 and let be a line bundle over g of degree 1 then the moduli space of rank 2 stable bundles v over g such that v was shown by seshadri 19 to be a nonsingular projective variety ng its rational cohomology ring h ng is exceedingly rich and despite years of study has never quite been computed in full the object of this paper is to give an essentially complete characterization of this ring or at least to reduce the problem to a matter of linear algebra in particular we find a proof of newstead s conjecture that pg1 ng 0 we also obtain a formula for the volume of ng which can be regarded as a twisted version of the formula for the degree 0 moduli space recently announced by witten the approach we shall take is not from algebraic geometry but from mathematical physics it relies on the su 2 wess zumino witten model which is a functor zk associating a finite dimensional vector space to each riemann surface with marked points the relationship with the moduli space is that when the level k of the functor is even the vector space associated to g with no marked points can be identified with h ng l where l is a fixed line bundle over ng now the work of verlinde provides us with a means of calculating the dimension of any vector space arising from our functor and in particular dimh ng l which we shall denote d g k on the other hand newstead 14 found explicit generators for h ng and we can also express d g k in terms of them using a riemann roch theorem equating the two formulas enables us to evaluate any monomial in the generators on the fundamental class of ng and by poincare duality this is sufficient at least in principle to determine the ring structure of h ng in the discussion above one important point has been skated over the su 2 wzw model is of course associated to bundles of degree 0 not degree 1 so in order to make use of verlinde s work it is necessary to formulate a twisted version of the field theory this is carried out in 2 but the crucial properties of the twisted theory analogous to those which make the ordinary theory a modular functor are not proved in this paper rather we will confine ourselves to exploring the consequences of these claims and hope to return to justify them in a later paper an outline of the remaining sections goes as follows in 3 we review those parts of verlinde s work we shall need show how they must be modified in the twisted case and work out some explicit formulas for d g k in 4 we study the cohomology of ng which we regard throughout as a space of representations via the theorem of narasimhan and seshadri we define newstead s generators and i of h ng

ordered regular equivalence relations on ordered semihypergroups abstract in this paper we introduce the concept of ordered regular strongly ordered regular equivalence relation on an ordered semihypergroup moreover we use hyperfilters to construct a strongly ordered regular equivalence relation on an ordered semihypergroup for which the corresponding quotient structure is a semilattice finally we construct an ordered regular equivalence relation on an ordered semihypergroup by hyperideals such that the corresponding quotient structure is also an ordered semihypergroup under a related order relation which is an answer to the open problem given by b davvaz p corsini and t changphas in european journal of combinatorics

isomorphism classes of the hypergroups of type u on the right of size five by means of a blend of theoretical arguments and computer algebra techniques we prove that the number of isomorphism classes of hypergroups of type u on the right of order five having a scalar bilateral identity is 14751 in this way we complete the classification of hypergroups of type u on the right of order five started in our preceding papers m de salvo d freni g lo faro a new family of hypergroups and hypergroups of type u on the right of size five far east j math sci 26 2 2007 393 418 m de salvo d freni g lo faro a new family of hypergroups and hypergroups of type u on the right of size five part two mathematicki vesnik 60 2008 23 45 m de salvo d freni g lo faro on the hypergroups of type u on the right of size five with scalar identity submitted for publication in particular we obtain that the number of isomorphism classes of such hypergroups is 14865

decomposability of linear maps under tensor powers both completely positive and completely copositive maps stay decomposable under tensor powers i e under tensoring the linear map with itself but are there other examples of maps with this property we show that this is not the case any decomposable map that is neither completely positive nor completely copositive will lose decomposability eventually after taking enough tensor powers moreover we establish explicit bounds to quantify when this happens to prove these results we use a symmetrization technique from the theory of entanglement distillation and analyze when certain symmetric maps become non decomposable after taking tensor powers finally we apply our results to construct new examples of non decomposable positive maps and establish a connection to the positive partial transpose squared conjecture

characterizations of strongly ordered regular relations on ordered semihypergroups abstract in this paper we give the fundamental normal homomorphism theorem of ordered semihypergroups moreover we characterize the strongly ordered regular relations on an ordered semihypergroup by means of the chain and obtain that the set of all strongly ordered regular relations on an ordered semihypergroup consists a complete lattice finally the subset of an ordered semihypergroup can serve as some ordered regular ordered semilattice relation class is determined