a note on set theoretic solutions of the yang baxter equation abstract this paper shows that every finite non degenerate involutive set theoretic solution x r of the yang baxter equation whose permutation group g x r has cardinality which is a cube free number is a multipermutation solution some properties of finite braces are also investigated it is also shown that if a is a left brace whose cardinality is an odd number and a b a b for all a b a then a is a two sided brace and hence a jacobson radical ring it is also observed that the semidirect product and the wreath product of braces of a finite multipermutation level is a brace of a finite multipermutation level

matched pairs approach to set theoretic solutions of the yang baxter equation we study set theoretic solutions x r of the yang baxter equations on a set x in terms of the induced left and right actions of x on itself we give a characterisation of involutive square free solutions in terms of cyclicity conditions we characterise general solutions in terms of abstract matched pair properties of the associated monoid s x r and we show that r extends as a solution rs on s x r as a set finally we study extensions of solutions both directly and in terms of matched pairs of their associated monoids we also prove several general results about matched pairs of monoids s of the required type including iterated products s s s equivalent to rs a solution and extensions s t rs t examples include a general double construction s s rs s and some concrete extensions their actions and graphs based on small sets

space time block codes from orthogonal designs we introduce space time block coding a new paradigm for communication over rayleigh fading channels using multiple transmit antennas data is encoded using a space time block code and the encoded data is split into n streams which are simultaneously transmitted using n transmit antennas the received signal at each receive antenna is a linear superposition of the n transmitted signals perturbed by noise maximum likelihood decoding is achieved in a simple way through decoupling of the signals transmitted from different antennas rather than joint detection this uses the orthogonal structure of the space time block code and gives a maximum likelihood decoding algorithm which is based only on linear processing at the receiver space time block codes are designed to achieve the maximum diversity order for a given number of transmit and receive antennas subject to the constraint of having a simple decoding algorithm the classical mathematical framework of orthogonal designs is applied to construct space time block codes it is shown that space time block codes constructed in this way only exist for few sporadic values of n subsequently a generalization of orthogonal designs is shown to provide space time block codes for both real and complex constellations for any number of transmit antennas these codes achieve the maximum possible transmission rate for any number of transmit antennas using any arbitrary real constellation such as pam for an arbitrary complex constellation such as psk and qam space time block codes are designed that achieve 1 2 of the maximum possible transmission rate for any number of transmit antennas for the specific cases of two three and four transmit antennas space time block codes are designed that achieve respectively all 3 4 and 3 4 of maximum possible transmission rate using arbitrary complex constellations the best tradeoff between the decoding delay and the number of transmit antennas is also computed and it is shown that many of the codes presented here are optimal in this sense as well

sum power iterative water filling for multi antenna gaussian broadcast channels in this correspondence we consider the problem of maximizing sum rate of a multiple antenna gaussian broadcast channel bc it was recently found that dirty paper coding is capacity achieving for this channel in order to achieve capacity the optimal transmission policy i e the optimal transmit covariance structure given the channel conditions and power constraint must be found however obtaining the optimal transmission policy when employing dirty paper coding is a computationally complex nonconvex problem we use duality to transform this problem into a well structured convex multiple access channel mac problem we exploit the structure of this problem and derive simple and fast iterative algorithms that provide the optimum transmission policies for the mac which can easily be mapped to the optimal bc policies

right l groups geometric garside groups and solutions of the quantum yang baxter equation abstract right lattice ordered groups are introduced and studied as a general framework for garside groups and related groups without a garside element every right l group g has a two sided partially ordered subgroup n g which generalizes the quasi centre of an artin tits group the group n g splits into copies of z if g is noetherian the positive cone of a right l group is described as a structure that is known from algebraic logic a pair of left and right self similar hoops for noetherian right l groups g modularity of the lattice structure is characterized in terms of an operation on the set x g of atoms it is proved that modular garside groups are equivalent to finite projective spaces with a non degenerate labelling a concept of duality for x g is introduced and applied in the distributive case this gives a one to one correspondence between noetherian right l groups with duality and non degenerate unitary set theoretic solutions of the quantum yang baxter equation the description of garside groups via garside germs is extended to right l groups which yields a one sided enhancement and a new proof of dvurecenskij s non commutative extension of mundici s correspondence between abelian l groups and mv algebras

affine structures of decomposable solvable groups abstract bi crossed products of groups are closely related to braces a ring like structure connected with solutions to the yang baxter equation the asymmetric product of braces which has become useful for the construction of regular affine groups and simple braces as well as several other product constructions are unified extended and simplified all bi crossed products of bicyclic braces are classified

multipermutation solutions of the yang baxter equation set theoretic solutions of the yang baxter equation form a meeting ground of mathematical physics algebra and combinatorics such a solution consists of a set x and a function r x x x x which satisfies the braid relation

on finite complete rewriting systems finite derivation type and automaticity for homogeneous monoids this paper investigates the class of finitely presented monoids defined by homogeneous length preserving relations from a computational perspective the properties of admitting a finite complete rewriting system having finite derivation type being automatic and being biautomatic are investigated for this class of monoids the first main result shows that for any consistent combination of these properties and their negations there is a homogeneous monoid with exactly this combination of properties we then introduce the new concept of abstract rees commensurability an analogue of the notion of abstract commensurability for groups in order to extend this result to show that the same statement holds even if one restricts attention to the class of n ary homogeneous monoids where every side of every relation has fixed length n we then introduce a new encoding technique that allows us to extend the result partially to the class of n ary multihomogenous monoids

low complexity user selection algorithms for multiuser mimo systems with block diagonalization block diagonalization bd is a precoding technique that eliminates interuser interference in downlink multiuser multiple input multiple output mimo systems with the assumptions that all users have the same number of receive antennas and utilize all receive antennas when scheduled for transmission the number of simultaneously supportable users with bd is limited by the ratio of the number of base station transmit antennas to the number of user receive antennas in a downlink mimo system with a large number of users the base station may select a subset of users to serve in order to maximize the total throughput the brute force search for the optimal user set however is computationally prohibitive we propose two low complexity suboptimal user selection algorithms for multiuser mimo systems with bd both algorithms aim to select a subset of users such that the total throughput is nearly maximized the first user selection algorithm greedily maximizes the total throughput whereas the criterion of the second algorithm is based on the channel energy we show that both algorithms have linear complexity in the total number of users and achieve around 95 of the total throughput of the complete search method in simulations

factorizations of skew braces we introduce strong left ideals of skew braces and prove that they produce non trivial decomposition of set theoretic solutions of the yang baxter equation we study factorization of skew left braces through strong left ideals and we prove analogs of ito s theorem in the context of skew left braces as a corollary we obtain applications to the retractability problem of involutive non degenerate solutions of the yang baxter equation finally we classify skew braces that contain no non trivial proper characteristic ideals

a covering theory for non involutive set theoretic solutions to the yang baxter equation abstract an algebraic theory of non involutive set theoretic solutions to the yang baxter equation by means of a non commutative version of braces is established the structure group and the permutation group of a solution is put into a context of discrete lie theory this leads to a natural concept of covering which shares the good properties of coverings in geometry and topology in the special case of quandle coverings in knot theory known anomalies are removed

on groups of i type and involutive yang baxter groups abstract we suggest a cohomological framework to describe groups of i type and involutive yang baxter groups these groups are key in the study of involutive non degenerate set theoretic solutions of the quantum yang baxter equation our main tool is a lifting criterion for 1 cocycles established here in a general non abelian setting

a decomposition theorem for square free unitary solutions of the quantum yang baxter equation abstract it is known that every skew polynomial ring with generating set x and binomial relations in the sense of gateva ivanova trans amer math soc 343 1994 203 is an artin schelter regular domain of global dimension x moreover every such ring gives rise to a non degenerate unitary set theoretical solution r x 2 x 2 of the quantum yang baxter equation which fixes the diagonal of x 2 gateva ivanova s conjecture talk at the international algebra conference miskolc hungary 1996 states that conversely every such solution r comes from a skew polynomial ring with binomial relations an equivalent conjecture duke math j 100 1999 169 says that the underlying set x is r decomposable we prove these conjectures and construct an indecomposable solution r with x which shows that an extension to infinite x is false

a combinatorial approach to the set theoretic solutions of the yang baxter equation a bijective map r x2 x2 where x x1 xn is a finite set is called a set theoretic solution of the yang baxter equation ybe if the braid relation r12r23r12 r23r12r23 holds in x3 a nondegenerate involutive solution x r satisfying r xx xx for all x x is called square free solution there exist close relations between the square free set theoretic solutions of ybe the semigroups of i type the semigroups of skew polynomial type and the bieberbach groups as it was first shown in a joint paper with michel van den bergh in this paper we continue the study of square free solutions x r and the associated yang baxter algebraic structures the semigroup s x r the group g x r and the k algebra a k x r over a field k generated by x and with quadratic defining relations naturally arising and uniquely determined by r we study the properties of the associated yang baxter structures and prove a conjecture of the present author that the three notions a square free solution of set theoretic ybe

the parameterization method in optimal control problems and differential algebraic equations the paper is devoted to the explanation of the numerical parameterization method pm for optimal control oc problems with intermediate phase constraint and to its circumstantiation for classical calculus of variation cv problems that arise in connection with singular odes or daes especially in cases of their essential degeneracy the pm is based on a finite parameterization of control functions and on derivation of the problem functional with respect to control parameters the first and the second derivatives are calculated with the help of adjoint vector and matrix impulses results of the solution to one phase constrained oc and two degenerate cv problems connected with singular daes nonreducible to the normal form are presented

finitely generated algebras defined by homogeneous quadratic monomial relations and their underlying monoids abstract we consider algebras over a field k with generators x 1 x 2 x n subject to n 2 quadratic relations of the form x i x j x k x l with i j k l and moreover every monomial x i x j appears at most once in one of the defining relations if these relations are non degenerate then it is shown that the algebra is left and right noetherian satisfies a polynomial identity and has gelfand kirillov dimension at most n in case the defining relations are square free this was already established by gateva ivanova jespers and okninski to prove these results we investigate the structure of the underlying monoid s defined by the same presentation it is called a quadratic monoid we show that there is a strong link with the divisibility monoids and monoids of i type also referred to as yb monoids monoids of i type are examples of non degenerate quadratic monoids they are a monoid interpretation of non degenerate involutive set theoretic solutions of the yang baxter equation and have been studied quite intensively in recent years by gateva ivanova and van den bergh etingof schedler and soloviev jespers and okninski divisibility monoids have been introduced by kuske these are cancellative monoids that include the class of monoids of i type we show that they have a presentation with at most n 2 relations and if they have precisely n 2 defining relations then they are monoids of i type

solving the yang baxter like matrix equation for a class of elementary matrices let u and v be two n dimensional vectors in the complex field such that v t u 0 and let a i u v t we find all the solutions of the quadratic matrix equation a x a x a x in various situations with the help of a spectral perturbation result for rank one updated matrices two numerical examples are given to demonstrate the effectiveness of the method including an application of the result to a 4 4 equation related to the classic yang baxter equation

complete commuting solutions of the yang baxter like matrix equation for diagonalizable matrices let a be a square matrix that is diagonalizable we find all the commuting solutions of the quadratic matrix equation a x a x a x by taking advantage of the jordan form structure of a together with the help of a well known theorem on the uniqueness of a solution to sylvester s equation two special classes of the given matrix a are further investigated including circular matrices and those that are equal to some of their powers moreover all the non commuting solutions are constructed when a is a householder matrix based on a spectral perturbation result

retractability of set theoretic solutions of the yang baxter equation abstract it is shown that square free set theoretic involutive non degenerate solutions of the yang baxter equation whose associated permutation group referred to as an involutive yang baxter group is abelian are retractable in the sense of etingof schedler and soloviev this solves a problem of gateva ivanova in the case of abelian iyb groups it also implies that the corresponding finitely presented abelian by finite groups called the structure groups are poly z groups secondly an example of a solution with an abelian involutive yang baxter group which is not a generalized twisted union is constructed this answers in the negative another problem of gateva ivanova the constructed solution is of multipermutation level 3 retractability of solutions is also proved in the case where the natural generators of the iyb group are cyclic permutations moreover it is shown that such solutions are generalized twisted unions

on commuting solutions of the yang baxter like matrix equation abstract let a be an arbitrary square matrix and its jordan canonical form is p 1 a p j diag j 1 1 j q q with p an invertible matrix 1 q are different eigenvalues of matrix a the commuting solution problem of the matrix equation a x a x a x is equivalent to the problem j i i y i y i j i i j i i 2 y i j i i y i 2 with y p 1 x p diag y 1 y q we give the structures of the commuting solutions y i in special toeplitz forms based on them we construct new matrices h 1 2 related to the commuting solutions then we propose a method of solving all the commuting yang baxter like solutions by which all solutions can be obtained step by step by recursively solving matrix equations in two cases i 0 or i 0 with respect to the i th jordan block j i i

numerical method for a class of optimal control problems subject to nonsmooth functional constraints in this paper we consider a class of optimal control problems which is governed by nonsmooth functional inequality constraints involving convolution first we transform it into an equivalent optimal control problem with smooth functional inequality constraints at the expense of doubling the dimension of the control variables then using the chebyshev polynomial approximation of the control variables we obtain an semi infinite quadratic programming problem at last we use the dual parametrization technique to solve the problem

quadratic algebras yang baxter equation and artin schelter regularity abstract we study two classes of quadratic algebras over a field k the class c n of all n generated pbw algebras with polynomial growth and finite global dimension and the class of quantum binomial algebras we show that a pbw algebra a is in c n iff its hilbert series is h a z 1 1 z n furthermore the class c n contains a unique up to isomorphism monomial algebra a k x 1 x n x j x i 1 i j n a surprising amount can be said when a is a quantum binomial algebra that is its defining relations are nondegenerate square free binomials x y c x y z t c x y k our main result shows that for an n generated quantum binomial algebra a the following conditions are equivalent i a is an artin schelter regular pbw algebra ii a is a yang baxter algebra that is the set of relations defines canonically a solution of the yang baxter equation iii a is a binomial skew polynomial ring with respect to an enumeration of x iv the koszul dual a is a quantum grassmann algebra

solutions of the yang baxter equation associated with a left brace abstract given a left brace b a method is given to construct explicitly all the non degenerate involutive set theoretic solutions x r of the yang baxter equation such that the associated permutation group g x r is isomorphic as a left brace to b this method depends entirely on the brace structure of b

optimality condition of the nonlinear impulsive system in fed batch fermentation abstract based on the nonlinear impulsive system of fed batch fermentation which is used in the process of bio dissimilation of glycerol to 1 3 propanediol by klebsiella pneumoniae this paper is concerned with the optimal control of the volumes of infused glycerol at impulsive moments the optimal control model is developed the authors study the properties of both the nonlinear impulsive system and the solution of it and ascertain the existence of the optimal control then the optimality condition is obtained which can be used as a necessary condition that the optimal control should satisfy

braces and the yang baxter equation several aspects of relations between braces and non degenerate involutive set theoretic solutions of the yang baxter equation are discussed and many consequences are derived in particular for each positive integer n a finite square free multipermutation solution of the yang baxter equation with multipermutation level n and an abelian involutive yang baxter group is constructed this answers a problem of gateva ivanova and cameron it is proved that finite non degenerate involutive set theoretic solutions of the yang baxter equation whose associated involutive yang baxter group is abelian are multipermutation solutions earlier the authors proved this with the additional square free hypothesis on the solutions it is also proved that finite square free non degenerate involutive set theoretic solutions associated to a left brace are multipermutation solutions

optimal control for nonlinear dynamical system of microbial fed batch culture in fed batch culture of glycerol bio dissimilation to 1 3 propanediol 1 3 pd the aim of adding glycerol is to obtain as much 1 3 pd as possible so a proper feeding rate is required during the process taking the concentration of 1 3 pd at the terminal time as the performance index and the feeding rate of glycerol as the control function we propose an optimal control model subject to a nonlinear dynamical system and constraints of continuous state and non stationary control a computational approach is constructed to seek the solution of the above model in two aspects on the one hand we transcribe the optimal control model into an unconstrained one based on the penalty functions and an extension of the state space on the other hand by approximating the control function with simple functions we transform the unconstrained optimal control problem into a sequence of nonlinear programming problems which can be solved using gradient based optimization techniques the convergence analysis of this approximation is also investigated numerical results show that by employing the optimal control policy the concentration of 1 3 pd at the terminal time can be increased considerably

writing on dirty paper corresp a channel with output y x s z is examined the state s sim n 0 qi and the noise z sim n 0 ni are multivariate gaussian random variables i is the identity matrix the input x in r n satisfies the power constraint l n sum i 1 n x i 2 leq p if s is unknown to both transmitter and receiver then the capacity is frac 1 2 ln 1 p n q nats per channel use however if the state s is known to the encoder the capacity is shown to be c ast frac 1 2 ln 1 p n independent of q this is also the capacity of a standard gaussian channel with signal to noise power ratio p n therefore the state s does not affect the capacity of the channel even though s is unknown to the receiver it is shown that the optimal transmitter adapts its signal to the state s rather than attempting to cancel it

involutive yang baxter groups in 1992 drinfeld posed the question of finding the set theoretic solutions of the yang baxter equation recently gateva ivanova and van den bergh and etingof schedler and soloviev have shown a group theoretical interpretation of involutive non degenerate solutions namely there is a one to one correspondence between involutive non degenerate solutions on finite sets and groups of i type a group g of i type is a group isomorphic to a subgroup of fa n sym n so that the projection onto the first component is a bijective map where fa n is the free abelian group of rank n and sym n is the symmetric group of degree n the projection of g onto the second component sym n we call an involutive yang baxter group iyb group this suggests the following strategy to attack drinfeld s problem for involutive non degenerate set theoretic solutions first classify the iyb groups and second for a given iyb group g classify the groups of i type with g as associated iyb group it is known that every iyb group is solvable in this paper some results supporting the converse of this property are obtained more precisely we show that some classes of groups are iyb groups we also give a non obvious method to construct infinitely many groups of i type and hence infinitely many involutive non degenerate set theoretic solutions of the yang baxter equation with a prescribed associated iyb group

set theoretic solutions of the yang baxter equation braces and symmetric groups abstract we involve simultaneously the theory of braided groups and the theory of braces to study set theoretic solutions of the yang baxter equation ybe we show the intimate relation between the notions of a symmetric group in the sense of takeuchi i e a braided involutive group and a left brace we find new results on symmetric groups of finite multipermutation level and the corresponding braces we introduce a new invariant of a symmetric group g r the derived chain of ideals of g which gives a precise information about the recursive process of retraction of g we prove that every symmetric group g r of finite multipermutation level m is a solvable group of solvable length m to each set theoretic solution x r of ybe we associate two invariant sequences of involutive braided groups i the sequence of its derived symmetric groups ii the sequence of its derived permutation groups and explore these for explicit descriptions of the recursive process of retraction of x r we find new criteria necessary and sufficient to claim that x r is a multipermutation solution

the maximum snr optimal weighting matrix for a class of amplify and forward mimo relaying assisted orthogonal space time block coded transmission the optimal weighting matrix design at the relay node for the v blast based amplify and forward af mimo relay transmission scheme has been intensively investigated in literature however there are few papers considering the optimal design of the weighting matrix at the relay node for orthogonal space time block codes ostbcs based af mimo relay transmission scheme in this paper the maximum snr weighting matrix at the relay node is designed for the ostbc based af mimo relay scheme so that for the first time we can quantify the snr gain by optimizing the weighting matrix at the relay node for the ostbc based af mimo relay scheme specifically the canonical form of the optimal linear weighting matrix is derived by exploiting matrix inequality theory and majorization theory furthermore we propose a simulation based gamma approximation method for diversity order analysis the simulation results show that the proposed optimal weighting matrix significantly outperforms the traditional scalar gain weighting matrix for the ostbc based af mimo relay scheme

semigroups ofi type abstract assume thatsis a semigroup generated by x1 xn and let u be the multiplicative free commutative semigroup generated by u1 un we say thatsis ofi typeif there is a bijectivev u ssuch that for alla u v u1a v una x1v a xnv a this condition appeared naturally in the work on sklyanin algebras by john tate and the second author in this paper we show that the condition for a semigroup to be ofi type is related to various other mathematical notions found in the literature in particular we show that semigroups ofi type appear in the study of the set theoretic solutions of the yang baxter equation in the theory of bieberbach groups and in the study of certain skew binomial polynomial rings which were introduced by the first author

commuting solutions of the yang baxter like matrix equation for a class of rank two updated matrices abstract let a i p q t where p and q are two n 2 complex matrices of full column rank and det q t p 0 all the commuting solutions with a of the quadratic matrix equation a x a x a x have been founded

end to end ber analysis for dual hop ostbc transmissions over rayleigh fading channels in this letter a ber study is presented for the end to end performance of dual hop wireless communication systems employing transmit diversity with orthogonal space time block codes ostbcs where a nonregenerative or regenerative relay is equipped with a single antenna operating over flat rayleigh fading channels more specifically we provide probability density functions pdfs and moment generating functions mgfs for the end to end snr of the dual hop ostbc transmissions and then present its ber performance over m ary qam and psk modulations respectively numerical investigation shows that the analytic ber provided in the letter makes an exact match with the simulation result in various multiple antenna transmission scenarios the result also shows how the number of antennas equipped at the source and destination affects the end to end performance