strict convexity of the feasible log sir region the feasible log sir region is defined as a set of all signal to interference ratios sir expressed in logarithmic scale that can be supported in a wireless network by means of power control and with all users being active concurrently recently the feasible log sir region was shown to be a convex set which is a key ingredient in the development of some power control strategies for wireless systems in this paper under the assumption of a noiseless channel we strengthen these results by proving a necessary and sufficient condition for the feasible log sir region to be a strictly convex set the strict convexity property is of interest since it is closely related to the problem of the existence and uniqueness of a so called log sir fair power vector

connections on discrete fibre bundles a new approach to gauge fields on a discrete space time is proposed in which the fundamental object is a discrete version of a principal fibre bundle if the bundle is twisted the gauge fields are topologically non trivial automatically

a calculus for log convex interference functions the behavior of certain interference coupled multiuser systems can be modeled by means of logarithmically convex log convex interference functions in this paper we show fundamental properties of this framework a key observation is that any log convex interference function can be expressed as an optimum over elementary log convex interference functions the results also contribute to a better understanding of certain quality of service qos tradeoff regions which can be expressed as sublevel sets of log convex interference functions we analyze the structure of the qos region and provide conditions for the achievability of boundary points the proposed framework of log convex interference functions generalizes the classical linear interference model which is closely connected with the theory of irreducible nonnegative matrices perron frobenius theory we discuss some possible applications in robust communication cooperative game theory and max min fairness

octagonal quasicrystals and a formula for shelling octagonal quasicrystals will be realized in a cyclotomic field and a formula for shelling will be given using number theory

hopf algebras of dimension pq ii abstract let h be a hopf algebra of dimension pq over an algebraically closed field of characteristic zero where p q are odd primes with p q 4 p 11 we prove that h is semisimple and thus isomorphic to a group algebra or the dual of a group algebra

classification of voronoi and delone tiles in quasicrystals i general method a new general method is presented which allows one to find all distinct voronoi and delone tiles in any quasicrystal from a large family this includes the tiles which may be present with arbitrarily low density 0 at all stages the method requires only consideration of a possibly large finite number of cases our method is applicable in principle to quasicrystals in any dimension and with any irrationality this is the first of three papers where the voronoi and delone tilings are studied two dimensional point sets quasicrystals arising from the a4 root lattice by means of the standard projection to a two dimensional plane with the irrationality 1 2 1 5 are considered in general we require that the acceptance window be bounded with non empty interior specific results are provided here for rhombic acceptance windows of any size oriented along the direction of simple roots of the coxeter group h2 within one quasicrystal the tiles are distinguished by their shape size and orientation the rhombic window case is indispensable for subsequent classification of voronoi and delone tiles in quasicrystals with general shape of the acceptance window voronoi and delone tiles of quasicrystals with circular and decagonal windows of any size are given in subsequent papers let vt denote the set of distinct voronoi tiles making up a quasicrystal with a given acceptance window there are three vt sets of the generic type and three of the singular type the latter occur for one precise value of the size of the acceptance window any other vt set is a uniform scaling of the tiles listed here similar results differing in detail are provided for the sets of distinct delone tiles dt altogether there are four different sets dt of delone tiles

on the even powers of the antipode of a finite dimensional hopf algebra abstract the trace of powers of the square of the antipode s 2 of a finite dimensional hopf algebra a over a field k is studied it is shown in many cases that the trace function vanishes on s 2 m when s 2 m 1 a finer properties of the antipode are related to this phenomenon

some topics on glq n abstract the quantum groups gl q n and sl q n are defined as pairs of hopf algebras and it is shown they are absolutely connected when q is a root of 1 some quantum analogue of the frobenius map and some related finite quantum groups are introduced and their simplicity is studied

combinatorial characterization of interference coupling in wireless systems we provide a combinatorial characterization of interference coupling in wireless systems with the intent of obtaining a better insight into interference coordination and management we introduce two bipartite graphs namely the power graph and interference graph we utilize these graphs and global dependency matrix containing only binary 0 and 1 entries to capture the effects of interference coupling in communication systems we show that the irreducibility of the global dependency matrix g is related to the connectivity of the power graph and the irreducibility of the matrix ggt is related to the connectivity of the interference graph we prove that for strictly positive and strictly log convex interference functions the irreducibility of the matrices g and ggt are necessary and sufficient conditions for the considered utility sets to be strictly convex in this case there exists a unique optimizer for the problem of maximizing the product of utilities we show that an interference balancing function is strictly log convex if and only if matrices g and ggt are irreducible we provide a simple yet comprehensive combinatorial characterization of interference coupled systems which abstracts away certain complexities of the physical layer

the set of types ofn dimensional semisimple and cosemisimple hopf algebras is finite in this paper we give a partial answer to the above statement proving that the set of types of semisimple and cosemisimple hopf algebras of a given dimension is finite in any characteristic the proof uses the formalism of deformation theory of bialgebras which is concerned from a geometrical point of view with the study of the affine algebraic set b v of all bialgebra structures which can be defined on a vector space v there are two basic facts regarding b v first of all the transport of structures defines an action of gl v on b v such that the orbit of a bialgebra a contains all structures isomorphic to a so the set of types of bialgebras on v is in a one to one correspondence with the set of orbits second we are led in a natural way to a cohomology theory for bialgebras studying the tangent spaces t o and t b v where o is the closure a a a a of the orbit through a bialgebra a one of the most important results of the deformation theory of bialgebras is that the orbit of a bialgebra is open if the cohomology vanishes therefore the number of orbits which correspond to bialgebras having trivial cohomology is finite we obtain our result by proving that the cohomology of a semisimple hopf algebra a vanishes if the characteristic of k is 0 or more generally if a is

the e8 lattice and quasicrystals geometry number theory and quasicrystals the authors study the quasiperiodic structures which can be derived from e8 this lattice suitably oriented leads to a 4d quasicrystal which has 3 3 5 symmetry they develop a modified version of the out and projection method the high dimensional lattice is first foliated into successive shells surrounding a vertex the shells are embedded into 7d s7 spheres they then use the so called hopf fibration of s7 to gather families of e8 shell sites into s3 fibres which contain 24 sites or a multiple of 24 which are symmetrically disposed in s3 this has two advantages first in the selection process a whole fibre is either selected or rejected which reduces the computation more importantly the selection process for a fiber amounts to simple arithmetical criteria the whole process leads to a shell by shell analysis of the 4d quasicrystal which can be brought onto a 2d 1d algorithm similar to the fibonacci chain construction they propose an arithmetic formula which gives the number of points on the shells this 4d quasicrystal has the interesting feature that it can be sliced into lower dimensional quasicrystals for example with icosahedral and tetrahedral symmetry in three dimensions

a generalization of nash bargaining and proportional fairness to log convex utility sets with power constraints many solutions and concepts in resource allocation and game theory rely on the assumption of a convex utility set in this paper we show that the less restrictive assumption of a logarithmic hidden convexity is sometimes sufficient we consider the problems of nash bargaining and proportional fairness which are closely related we extend the nash bargaining framework to a broader family of log convex sets we then focus on the set of feasible signal to interference plus noise ratios sinrs for the cases of individual power constraints and a sum power constraint under the assumption of log convex interference functions we show how pareto optimality of boundary points depends on the interference coupling between the users finally we provide necessary and sufficient conditions for strict log convexity of the feasible sinr region

a freeness theorem for nichols algebras abstract we prove a freeness result for nichols algebras over certain nichols subalgebras this result allows us in particular to classify pointed hopf algebras of prime and prime squared index

capacity regions and optimal power allocation for cdma cellular radio the capacity region of code division multiple access cdma is determined by the set of transmission rates combined with quality of service qos requirements which allow for a feasible power allocation scheme for n mobiles in a cellular network the geometrical and topological properties of the capacity region are investigated in the present paper for the case of unlimited and limited power respectively as a central result we show that the capacity region is convex by breaking the complicated topological structure into characteristic properties of its boundary and interior points each of interest in itself based on these results we furthermore investigate optimal power assignment schemes in the case that the demand of a community of users is infeasible weighted minimax and bayes solutions are explicitly determined as appropriate means to share the capacity of a cellular network in a reasonable and fair way

a sharpened condition for strict log convexity of the spectral radius via the bipartite graph abstract friedland 1981 10 showed that for a nonnegative square matrix a the spectral radius r e d a is a log convex functional over the real diagonal matrices d he showed that for fully indecomposable a log r e d a is strictly convex over d 1 d 2 if and only if d 1 d 2 c i for any c r here the condition of full indecomposability is shown to be replaceable by the weaker condition that a and a a be irreducible which is the sharpest possible replacement condition irreducibility of both a and a a is shown to be equivalent to irreducibility of a 2 and a a which is the condition for a number of strict inequalities on the spectral radius found in cohen et al 1982 8 such two fold irreducibility is equivalent to joint irreducibility of a a 2 a a and aa or in combinatorial terms equivalent to the directed graph of a being strongly connected and the simple bipartite graph of a being connected additional ancillary results are presented

quasicrystals and icosians a family of quasicrystals of dimensions 1 2 3 4 governed by the root lattice e8 is constructed the use of the icosian ring found in the quaternions with coefficients in q square root 5 allows simultaneous interpretation of the construction both in physical space and as a result of the standard cut and projection method in double dimension icosians are seen to provide a natural co ordination scheme for these quasicrystals nested sequences of quasicrystals form systems whose symmetries are all derivable from inflational and reflective symmetries directly related to the arithmetic of the icosians the use of coxeter diagrams clarifies the amazing relationship of e8 and quasicrystal symmetries and leads to the fundamental chain e8 contains implies d6 contains implies a4 contains implies a1 a1 that underlies five fold symmetry in quasicrystals decomposition of quasicrystals into concentric shells and a counting formula for the cardinalities of these shells is discussed

a classification result on prime hopf algebras of gk dimension one abstract in this paper we classify all prime hopf algebras h of gk dimension one satisfying the following two conditions 1 h has a 1 dimensional representation of order pi deg h and 2 the invariant components of h with respect to this 1 dimensional representation are domains see section 2 for related definitions as consequences 1 a number of new hopf algebras of gk dimension one are found and some of them are not pointed 2 we give a partial answer to a question posed in 12 and 3 the question 1 1 is solved

lifting of quantum linear spaces and pointed hopf algebras of orderp3 abstract we propose the following principle to study pointed hopf algebras or more generally hopf algebras whose coradical is a hopf subalgebra given such a hopf algebraa consider its coradical filtration and the associated graded coalgebra gr a then gr ais a graded hopf algebra since the coradicala0ofais a hopf subalgebra in addition there is a projection gr a a0 letrbe the algebra of coinvariants of then by a result of radford and majid ris a braided hopf algebra and gr ais the bosonization or biproduct ofranda0 gr a r a0 the principle we propose to studyais first to studyr then to transfer the information to gr avia bosonization and finally to lift toa in this article we apply this principle to the situation whenris the simplest braided hopf algebra a quantum linear space as consequences of our technique we obtain the classification of pointed hopf algebras of orderp3 pan odd prime over an algebraically closed field of characteristic zero with the same hypothesis the characterization of the pointed hopf algebras whose coradical is abelian and has indexporp2 and an infinite family of pointed nonisomorphic hopf algebras of the same dimension this last result gives a negative answer to a conjecture of i kaplansky

hochschild cohomology and the coradical filtration of pointed coalgebras applications abstract in this paper we show that there is a close connection between the coradical filtration of a pointed coalgebra and the hochschild cohomology of that coalgebra with coefficients in some one dimensional bicomodules as an application for a given prime numberpand an algebraically closed fieldkof characteristic 0 we classify all pointed hopf algebras of dimensionp3overk

classification of arithmetic root systems arithmetic root systems are invariants of nichols algebras of diagonal type with a certain finiteness property they can also be considered as generalizations of ordinary root systems with rich structure and many new examples on the other hand nichols algebras are fundamental objects in the construction of quantized enveloping algebras in the noncommutative differential geometry of quantum groups and in the classification of pointed hopf algebras by the lifting method of andruskiewitsch and schneider in the present paper arithmetic root systems are classified in full generality as a byproduct many new finite dimensional pointed hopf algebras are obtained

cross products by braided groups and bosonization let h be a braided cocommutative hopf algebra in a braided monoidal category c and b a hopf algebra in c on which h acts we construct a cross product hopf algebra b formula h in c as an application we show that every b in a certain class can be converted to an equivalent ordinary hopf algebra by a process of bosonization the class includes for example all super hopf algebras the constructions respect any quasitriangular structures on b as a corollary we show that if h r is an ordinary quasitriangular hopf algebra then the smash product by the adjoint action of h on itself had formula h can be given the structure of a hopf algebra we prove a similar result for any hopf algebra to which a quasitriangular hopf algebra maps

on the number of types of finite dimensional hopf algebras for an odd prime p we construct an infinite class of non isomorphic hopf algebras of dimension p 4 over an infinite field containing primitive p th roots of unity answering in the negative a long standing conjecture of kaplansky

cocycle deformations for hopf algebras with a coalgebra projection abstract let h be a hopf algebra over a field k of characteristic 0 and let a be a bialgebra or hopf algebra such that h is isomorphic to a sub hopf algebra of a and there is an h bilinear coalgebra projection from a to h which splits the inclusion then a r h where r is the pre bialgebra of coinvariants in this paper we study the deformations of a by an h bilinear cocycle if is a cocycle for a then can be restricted to a cocycle r for r and a r r h as examples we consider liftings of b v k where is a finite abelian group v is a quantum plane and b v is its nichols algebra and explicitly construct the cocycle which twists the radford biproduct into the lifting

root lattices and quasicrystals it is shown that root lattices and their reciprocals might serve as the right pool for the construction of quasicrystalline structure models all noncrystallographic symmetries observed so far are covered in minimal embedding with maximal symmetry

pointed hopf algebras of dimensionp3 abstract we give a structure theorem for pointed hopf algebras of dimension p 3 having coradical kc p where k is an algebraically closed field of characteristic zero combining this with previous results we obtain the complete classification of all pointed hopf algebras of dimension p 3

a monoidal approach to splitting morphisms of bialgebras the main goal of this paper is to investigate the structure of hopf algebras with the property that either its jacobson radical is a hopf ideal or its coradical is a subalgebra let us consider a hopf algebra a such that its jacobson radical j is a nilpotent hopf ideal and h a j is a semisimple algebra we prove that the canonical projection of a on h has a section which is an h colinear algebra map furthermore if h is cosemisimple too then we can choose this section to be an h h bicolinear algebra morphism this fact allows us to describe a as a generalized bosonization of a certain algebra r in the category of yetter drinfeld modules over h as an application we give a categorical proof of radford s result about hopf algebras with projections we also consider the dual situation let a be a bialgebra such that its coradical is a hopf sub bialgebra with antipode then there is a retraction of the canonical injection of h into a which is an h linear coalgebra morphism furthermore if h is semisimple too then we can choose this retraction to be an h h bilinear coalgebra morphism then also in this case we can describe a as a generalized bosonization of a certain coalgebra r in the category of yetter drinfeld modules over h

a highly symmetric four dimensional quasicrystal a quasiperiodic pattern or quasicrystal is constructed in real four dimensional euclidean space having the non crystallographic reflection group 3 3 5 of order 14 400 as its point group it is obtained as a projection of the eight dimensional lattice e8 and has as a cross section a three dimensional quasicrystal with icosahedral symmetry

on hopf algebras of dimension 4p abstract in this paper we prove that a non semisimple hopf algebra h of dimension 4 p with p an odd prime over an algebraically closed field of characteristic zero is pointed provided h contains more than two group like elements in particular we prove that non semisimple hopf algebras of dimensions 20 28 and 44 are pointed or their duals are pointed and this completes the classification of hopf algebras in these dimensions

the structure of hopf algebras with a projection the smash product algebra and the smash coproduct coalgebra are well known in the context of hopf algebras 1 5 and these notions can be viewed as being motivated by the semidirect product construction in the theory of groups and in the theory of afiine group schemes respectively let

finite dimensional cosemisimple hopf algebras in characteristic 0 are semisimple finite dimensional cosemisimple hopf algebras are a natural generalization of group algebras much of the representation theory of finite groups can be extended to such hopf algebras however there are still many unanswered basic questions for such hopf algebras one such question is a conjecture mentioned by kaplansky i the square of the antipode of a finite dimensional cosemisimple hopf algebra is the identity in this paper we prove that the fourth power of the antipode is the identity in a finite dimensional cosemisimple hopf algebra over a field of characteristic 0 in the course of proving this we investigate two elements of the hopf algebra which are roughly analogous to the sum of the group elements in a group algebra we show that kaplansky s conjecture is equivalent to these two elements being equal for a finite dimensional hopf algebra a with antipode s over a field let l p q pq for p q e a denote the left module action of a on itself in this paper we study two elements of a defined by the trace function the element i defined by p a tr l p s for all pe a and the element x defined by p x tr l p for all p e a we show that 2 is a left integral and use it to prove theorem 3 3 the main result of this paper a finite dimensional cosemisimple hopf algebra over a field of characteristic 0 is also semisimple hence its antipode has order 1 2 or 4 we extend the first corollary to 3 proposition 91 in theorems 4 3 and 4 4 by finding necessary and sufficient conditions for s to be a non zero left integral in which case x a and also study how x relates to the structure of a in particular when dim a 1 0 the right ideal xa may be thought of as a measure of the extent to which s i the elements ji and k are two examples of elements of a associated with an endomorphism of a in general for f end a we define a e a by p a tr l p cf for all pea thus i as2 and x a in section 2 267 0021 8693 88 33 00

on a factorization of graded hopf algebras using lyndon words abstract we find a generalization of the restricted pbw basis for pointed hopf algebras over abelian groups constructed by kharchenko we obtain a factorization of the hilbert series for a wide class of graded hopf algebras these factors are parametrized by lyndon words and they are the hilbert series of certain graded hopf algebras