hopf algebras having a dense big cell we discuss some axioms that ensure that a hopf algebra has its simple comodules classified using an analogue of the borel weil construction more precisely we show that a hopf algebra having a dense big cell satisfies the above requirement this method has its roots in the work of parshall and wang in the case of q deformed quantum groups gl and sl here we examine the example of universal cosovereign hopf algebras for which the weight group is the free group on two generators

the dominguez lorenzo condition and multivalued nonexpansive mappings abstract let e be a nonempty bounded closed convex separable subset of a reflexive banach space x which satisfies the dominguez lorenzo condition i e an inequality concerning the asymptotic radius of a sequence and the chebyshev radius of its asymptotic center we prove that a multivalued nonexpansive mapping t e 2 x which is compact convex valued and such that t e is bounded and satisfies an inwardness condition has a fixed point as a consequence we obtain a fixed point theorem for multivalued nonexpansive mappings in uniformly nonsquare banach spaces which satisfy the property worth extending a known result for the case of nonexpansive single valued mappings we also prove a common fixed point theorem for two nonexpansive commuting mappings t e e and t e kc e where kc e denotes the class of all compact convex subsets of e when x is a uniformly convex banach space

fixed point theorems for multivalued nonexpansive mappings satisfying inwardness conditions abstract let x be a banach space whose characteristic of noncompact convexity is less than 1 and satisfies the nonstrict opial condition let c be a bounded closed convex subset of x kc x the family of all compact convex subsets of x and t a nonexpansive mapping from c into kc x with bounded range we prove that t has a fixed point the nonstrict opial condition can be removed if in addition t is an 1 contractive mapping

nested lattice codes for gaussian relay networks with interference in this paper we consider a class of single source multicast relay networks we assume that all outgoing channels of a node in the network to its neighbors are orthogonal while the incoming signals from its neighbors can interfere with each other we first focus on gaussian relay networks with interference and find an achievable rate using a lattice coding scheme we show that the achievable rate of our scheme is within a constant bit gap from the information theoretic cut set bound where the constant depends only on the network topology but not on the transmit power noise variance and channel gains this is similar to a recent result by avestimehr diggavi and tse who showed an approximate capacity characterization for general gaussian relay networks however our achievability uses a structured code instead of a random one using the idea used in the gaussian case we also consider a linear finite field symmetric network with interference and characterize its capacity using a linear coding scheme

free products of compact quantum groups we construct and study compact quantum groups from free products ofc algebras in this connection we discover two mysterious classes of natural compact quantum groups a u m anda o m thea u m s respectivelya o m s are non isomorphic to each other for differentm s and are not obtainable by the ordinary quantization method we also clarify some basic concepts in the theory of compact quantum groups

grothendieck rings of universal quantum groups abstract we determine the grothendieck ring of finite dimensional comodules for the free hopf algebra on a matrix coalgebra and similarly for the free hopf algebra with bijective antipode and other related universal quantum groups the results turn out to parallel those for wang and van daele s universal compact quantum groups and bichon s generalization of those results to universal cosovereign hopf algebras in all cases the rings are isomorphic to those of non commutative polynomials over certain sets these sets varying from case to case in most cases we are able to give more precise information about the multiplication table of the grothendieck ring

quantum subgroups of glα β n abstract let c 0 and l n be odd with l 3 we determine all hopf algebra quotients of the quantized coordinate algebra o gl n when 1 is a primitive l th root of unity and satisfy certain mild conditions and we characterize all finite dimensional quotients when 1 is not a root of unity as a byproduct we find a new family of non semisimple and non pointed hopf algebras with non pointed duals which are quotients of o gl n

a network information theory for wireless communication scaling laws and optimal operation how much information can be carried over a wireless network with a multiplicity of nodes and how should the nodes cooperate to transfer information to study these questions we formulate a model of wireless networks that particularly takes into account the distances between nodes and the resulting attenuation of radio signals and study a performance measure that weights information by the distance over which it is transported consider a network with the following features i n nodes located on a plane with minimum separation distance spl rho sub min 0 ii a simplistic model of signal attenuation e sup spl gamma spl rho spl rho sup spl delta over a distance spl rho where spl gamma spl ges 0 is the absorption constant usually positive unless over a vacuum and spl delta 0 is the path loss exponent iii all receptions subject to additive gaussian noise of variance spl sigma sup 2 the performance measure we mainly but not exclusively study is the transport capacity c sub t sup spl sigma on sub spl lscr 1 sup m r sub spl lscr spl middot spl rho sub spl lscr where the supremum is taken over m and vectors r sub 1 r sub 2 r sub m of feasible rates for m source destination pairs and spl rho sub spl lscr is the distance between the spl lscr th source and its destination it is the supremum distance weighted sum of rates that the wireless network can deliver we show that there is a dichotomy between the cases of relatively high and relatively low attenuation when spl gamma 0 or spl delta 3 the relatively high attenuation case the transport capacity is bounded by a constant multiple of the sum of the transmit powers of the nodes in the network however when spl gamma 0 and spl delta 3 2 the low attenuation case we show that there exist networks that can provide unbounded transport capacity for fixed total power yielding zero energy priced communication examples show that nodes can profitably cooperate over large distances using coherence and multiuser estimation when the attenuation is low these results are established by developing a coding scheme and an achievable rate for gaussian multiple relay channels a result that may be of interest in its own right

joint physical layer coding and network coding for bidirectional relaying we consider a communication system where two transmitters wish to exchange information through a central relay the transmitter and relay nodes exchange data over synchronized average power constrained additive white gaussian noise channels with a real input with signal to noise ratio snr of snr an upper bound on the capacity is 1 2 log 1 snr bits per transmitter per use of the multiple access phase and broadcast phase of the bidirectional relay channel we show that using lattice codes and lattice decoding we can obtain a rate of 1 2 log 1 2 snr bits per transmitter which is essentially optimal at high snr the main idea is to decode the sum of the codewords modulo a lattice at the relay followed by a broadcast phase which performs slepian wolf coding we also show that if the two transmitters use identical lattices with minimum angle decoding we can achieve the same rate of 1 2 log 1 2 snr the proposed scheme can be thought of as a joint physical layer network layer code which outperforms other recently proposed analog network coding schemes

the jordan von neumann constants and fixed points for multivalued nonexpansive mappings the purpose of this paper is to study the existence of fixed points for nonexpansive multivalued mappings in a particular class of banach spaces furthermore we demonstrate a relationship between the weakly convergent sequence coefficient wcs x and the jordan von neumann constant cnj x of a banach space x using this fact we prove that if cnj x is less than an appropriate positive number then every multivalued nonexpansive mapping t e kc e has a fixed point where e is a nonempty weakly compact convex subset of a banach space x and kc e is the class of all nonempty compact convex subsets of e

borel weil theorem for algebraic supergroups abstract we study the structure of an algebraic supergroup and establish the borel weil theorem for to give a systematic construction of all simple supermodules over an arbitrary field especially when has a distinguished parabolic super subgroup we show that the set of all simple supermodules of is parameterized by the set of all dominant weights for the even part of prove a super analogue of the kempf vanishing theorem and give a description of euler characteristics

fixed point theorems for some generalized multivalued nonexpansive mappings abstract in this paper we introduce a condition on multivalued mappings which is a multivalued version of condition c defined by garcia falset et al 2011 3 it is shown here that some of the classical fixed point theorems for multivalued nonexpansive mappings can be extended to mappings satisfying this condition our results generalize the results in lim 1974 lami dozo 1973 kirk and massa 1990 garcia falset et al 2011 dhompongsa et al 2009 and abkar and eslamian 2010 4 5 6 3 7 8 and many others

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

quantum symmetry groups of finite spaces we determine the quantum automorphism groups of finite spaces these are compact matrix quantum groups in the sense of woronowicz

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

a noncommutative de finetti theorem invariance under quantum permutations is equivalent to freeness with amalgamation we show that the classical de finetti theorem has a canonical noncommutative counterpart if we strengthen exchangeability i e invariance of the joint distribution of the random variables under the action of the permutation group to invariance under the action of the quantum permutation group more precisely for an infinite sequence of noncommutative random variables x i i in mathbb n we prove that invariance of the joint distribution of the xi s under quantum permutations is equivalent to the fact that the xi s are identically distributed and free with respect to the conditional expectation onto the tail algebra of the xi s

integration over quantum permutation groups abstract we find a combinatorial formula for the haar measure of quantum permutation groups this leads to a dynamic formula for laws of diagonal coefficients explaining the poisson free poisson convergence result for characters

on cotriangular hopf algebras in an earlier paper we proved that any triangular semisimple hopf algebra over an algebraically closed field k of characteristic 0 is obtained from the group algebra k g of a finite group g by twisting its comultiplication by a twist in the sense of drinfeld in this paper we generalize this result to not necessarily finite dimensional cotriangular hopf algebras namely our main result says that a cotriangular hopf algebra a over k is obtained from a function algebra of a proalgebraic group by twisting its multiplication by a hopf 2 cocycle and possibly changing its r form by a central grouplike element of a of order 2 if and only if the trace of the squared antipode on any finite dimensional subcoalgebra of a is the dimension of this subcoalgebra the generalization like the original theorem is proved using deligne s theorem on tannakian categories we then give examples of twisted function algebras and in particular show that in the infinite dimensional case the squared antipode may not equal the identity on the other hand we show that in all of our examples the squared antipode is unipotent and conjecture it to be the case for any twisted function algebra we prove this conjecture in a large number of special cases using the quantization theory of the first author and d kazhdan

compute and forward harnessing interference through structured codes interference is usually viewed as an obstacle to communication in wireless networks this paper proposes a new strategy compute and forward that exploits interference to obtain significantly higher rates between users in a network the key idea is that relays should decode linear functions of transmitted messages according to their observed channel coefficients rather than ignoring the interference as noise after decoding these linear equations the relays simply send them towards the destinations which given enough equations can recover their desired messages the underlying codes are based on nested lattices whose algebraic structure ensures that integer combinations of codewords can be decoded reliably encoders map messages from a finite field to a lattice and decoders recover equations of lattice points which are then mapped back to equations over the finite field this scheme is applicable even if the transmitters lack channel state information

insufficiency of linear coding in network information flow it is known that every solvable multicast network has a scalar linear solution over a sufficiently large finite field alphabet it is also known that this result does not generalize to arbitrary networks there are several examples in the literature of solvable networks with no scalar linear solution over any finite field however each example has a linear solution for some vector dimension greater than one it has been conjectured that every solvable network has a linear solution over some finite field alphabet and some vector dimension we provide a counterexample to this conjecture we also show that if a network has no linear solution over any finite field then it has no linear solution over any finite commutative ring with identity our counterexample network has no linear solution even in the more general algebraic context of modules which includes as special cases all finite rings and abelian groups furthermore we show that the network coding capacity of this network is strictly greater than the maximum linear coding capacity over any finite field exactly 10 greater so the network is not even asymptotically linearly solvable it follows that even for more general versions of linearity such as convolutional coding filter bank coding or linear time sharing the network has no linear solution

on the classification of finite dimensional pointed hopf algebras we classify finite dimensional complex hopf algebras a which are pointed that is all of whose irreducible comodules are one dimensional and whose group of group like elementsg a is abelian such that all prime divisors of the order of g a are 7 since these hopf algebras turn out to be deformations of a natural class of generalized small quantum groups our result can be read as an axiomatic description of generalized small quantum groups

liberation of orthogonal lie groups abstract we show that under suitable assumptions we have a one to one correspondence between classical groups and free quantum groups in the compact orthogonal case we classify the groups under correspondence with the result that there are exactly 6 of them o n s n h n b n s n b n we investigate the representation theory aspects of the correspondence with the result that for o n s n h n b n this is compatible with the bercovici pata bijection finally we discuss some more general classification problems in the compact orthogonal case notably with the construction of a new quantum group

line and lattice networks under deterministic interference models capacity bounds are compared for four different deterministic models of wireless networks representing four different ways of handling broadcast and superposition in the physical layer in particular the transport capacity under a multiple unicast traffic pattern is studied for a 1 d network of regularly spaced nodes on a line and for a 2 d network of nodes placed on a hexagonal lattice the considered deterministic models are i p p a model with exclusive transmission and reception ii p m a model with simultaneous reception of the sum of the signals transmitted by all nearby nodes iii b p a model with simultaneous transmission to all nearby nodes but exclusive reception and iv b m a model with both simultaneous transmission and simultaneous reception all four deterministic models are considered under half duplex constraints for the 1 d scenario it is found that the transport capacity under b m is twice that under p p for the 2 d scenario it is found that the transport capacity under b m is at least 2 5 times and no more than six times the transport capacity under p p the transport capacities under p m and b p fall between these bounds

fixed point theorems for nonexpansive mappings and suzuki generalized nonexpansive mappings on a banach lattice abstract we prove the existence of fixed points for multivalued nonexpansive nonself mappings on a weakly orthogonal reflexive banach lattice with uniformly monotone norm moreover for single valued mappings we extend betiuk pilarska and prus s result a betiuk pilarska s prus banach lattices which are order uniformly noncreasy j math anal appl 342 2008 1271 1279 on the weak fixed point property to continuous mappings satisfying condition c on a w weakly orthogonal ounc banach lattice

a fixed point theorem for pointwise eventually nonexpansive mappings in nearly uniformly convex banach spaces abstract the purpose of this paper is to prove the existence of a fixed point for a pointwise eventually nonexpansive mapping in a nearly uniformly convex banach space this provides an affirmative answer to a question given by kirk and xu w a kirk hong kun xu asymptotic pointwise contraction nonlinear anal 69 2008 4706 4712

the james constant the jordan von neumann constant weak orthogonality and fixed points for multivalued mappings abstract we give some sufficient conditions for the dominguez lorenzo condition in terms of the james constant the jordan von neumann constant and the coefficient of weak orthogonality as a consequence we obtain fixed point theorems for multivalued nonexpansive mappings

fixed points of weakly inward multis soit x un espace de banach reel k x convexe ferme borne f k 2 x semicontinue superieurement et condensant f x convexe ferme x k et f faiblement interieur sur k alors f a un point fixe

banach ultrapowers and multivalued nonexpansive mappings the banach ultrapower construction is applied in fixed point theory for multivalued mappings we introduce the notion of ultra asymptotic centers and use it to remove the separability assumption from the results of dominguez benavides lorenzo ramirez 2004 and dhompongsa kaewcharoen kaewkhao 2006

de finetti theorems for easy quantum groups we study sequences of noncommutative random variables which are invariant under quantum transformations coming from an orthogonal quantum group satisfying the easiness condition axiomatized in our previous paper for 10 easy quantum groups we obtain de finetti type theorems characterizing the joint distribution of any infinite quantum invariant sequence in particular we give a new and unified proof of the classical results of de finetti and freedman for the easy groups sn on which is based on the combinatorial theory of cumulants we also recover the free de finetti theorem of kostler and speicher and the characterization of operator valued free semicircular families due to curran we consider also finite sequences and prove an approximation result in the spirit of diaconis and freedman