the racah wigner approach to harmonic oscillations of the cluster of nodes of a cube the racah wigner approach is applied to demonstrate various group theoretic labelling schemes for modes of oscillations and elasticity parameters of the cluster of nodes of a cube it is shown that there exists a fundamental basis in the configuration space closed under the action of the group oh the geometric symmetry group of the equilibrium it provides a transparent interpretation of all basic invariants the fibre bundle and tensor product structure of invariant bilinear forms over the configuration space yields three classification schemes for elasticity parameters in a way that resembles racah recoupling of several angular momenta

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

classical solutions of the quantum yang baxter equation the classical analogue is developed here for part of the construction in which knot and link invariants are produced from representations of quantum groups whereas previous work begins with a quantum group obtained by deforming the multiplication of functions on a poisson lie group we work directly with a poisson lie groupg and its associated symplectic groupoid the classical analog of the quantumr matrix is a lagrangian submanifold open image in new window in the cartesian square of the symplectic groupoid for any symplectic leafs ing open image in new window induces a symplectic automorphism ofs s which satisfies the set theoretic yang baxter equation when combined with the flip map exchanging components and suitably implanted in each cartesian powers n generates a symplectic action of the braid groupb n ons n application of a symplectic trace formula to the fixed point set of the action of braids should lead to link invariants but work on this last step is still in progress

symmetry properties of direct sum systems an interpretation of double coset decompositions of groups is given in terms of the construction of systems from similar disjoint subsystems a natural labelling scheme is developed for the direct product states of systems of this type providing an extension of the concept of unique labelling an application of the decoupling transformation to finite direct sum systems is investigated

skew braces and hopf galois structures of heisenberg type abstract we classify all skew braces of heisenberg type for a prime number p 3 furthermore we determine the automorphism group of each one of these skew braces as well as their socle and annihilator hence by utilising a link between skew braces and hopf galois theory we can determine all hopf galois structures of heisenberg type on galois field extensions of fields of degree p 3

on the set theoretical yang baxter equation we propose a general way of constructing set theoretical solutions of the yangbaxter equation we study the properties of the construction we also show that our construction includes the earlier ones given by weinstein xu and etingof schedlersoloviev

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

a chebyshev expansion method for solving nonlinear optimal control problems this paper presents a numerical technique for solving the controlled duffing oscillator the control and state variables are approximated by chebyshev series a technique is provided for approximating the system dynamics boundary conditions and performance index this technique is based on using an explicit formula for the chebyshev polynomials in terms of arbitrary order of their derivatives the system dynamics and the performance index are converted into some algebraic equations then the optimal control problem is reduced to constrained optimization problem results and comparisons are given at the end of this paper

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

set theoretic solutions of the yang baxter equation and new classes of r matrices abstract we describe several methods of constructing r matrices that are dependent upon many parameters for example unitary r matrices and r matrices whose entries are functions as an application we construct examples of r matrices with prescribed singular values we characterise some classes of indecomposable set theoretic solutions of the quantum yang baxter equation qybe and construct r matrices related to such solutions in particular we establish a correspondence between one generator braces and indecomposable non degenerate involutive set theoretic solutions of the qybe showing that such solutions are abundant we show that r matrices related to involutive non degenerate solutions of the qybe have special form we also investigate some linear algebra questions related to r matrices

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

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

unique labelling schemes for local modes in lattices two procedures are described which together specify a unique point group labelling scheme for the irreducible components of local distortions in crystals in the first procedure the group operators are factorised into a permutation part and a local atomic point group operation the second depends on dividing the complete shells of atoms into subshells which are invariant under subgroups of the site permutation group

a chebyshev technique for solving nonlinear optimal control problems a numerical technique for solving nonlinear optimal control problems is introduced the state and control variables are expanded in the chebyshev series and an algorithm is provided for approximating the system dynamics boundary conditions and performance index application of this method results in the transformation of differential and integral expressions into systems of algebraic or transcendental expressions in the chebyshev coefficients the optimum condition is obtained by applying the method of constrained extremum for linear quadratic optimal control problems the state and control variables are determined by solving a set of linear equations in the chebyshev coefficients applicability is illustrated with the minimum time and maximum radius orbit transfer problems

the interplay between k graphs and the yang baxter equation in this paper we initiate the study of the interplay between k graphs and the yang baxter equation for this we provide two very different perspectives on one hand we show that the set of all set theoretic solutions of the yang baxter equation is a special class of single vertex k graphs as a consequence we construct an infinite family of large solutions of the yang baxter equation from an arbitrarily given one on the other hand we prove that all single vertex k graphs are yb semigroups of square free involutive solutions of the yang baxter equation other various connections are also investigated

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

semi braces and the yang baxter equation abstract in this paper we obtain new solutions of the yang baxter equation that are left non degenerate through left semi braces a generalization of braces introduced by rump in order to provide new solutions we introduce the asymmetric product of left semi braces a generalization of the semidirect product of braces that allows us to produce several examples of left semi braces

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

numerical solution of the controlled duffing oscillator by hybrid functions a numerical method for solving the controlled duffing oscillator is presented the method is based upon hybrid functions approximation the properties of hybrid functions which consists of block pulse functions plus legendre polynomials are given the method is based upon expanding various time functions in the system as their truncated hybrid functions the operational matrix of product is introduced this matrix together with the operational matrix of integration are utilized to reduce the solution of controlled duffing oscillator to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique

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

a numerical solution of the nonlinear controlled duffing oscillator by radial basis functions in this research a new numerical method is applied to investigate the nonlinear controlled duffing oscillator this method is based on the radial basis functions rbfs to approximate the solution of the optimal control problem by using the collocation method we apply legendre gauss lobatto points for rbfs center nodes in order to use the numerical integration method more easily then the method of lagrange multipliers is used to obtain the optimum of the problems for this purpose different applications of rbfs are used the differential and integral expressions which arise in the dynamic systems the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients illustrative examples are included to demonstrate the validity and applicability of the technique

on the transition from regular to chaotic behaviour in the duffing oscillator a numerical procedure is introduced to study the transition from regular periodic motion to chaotic behaviour in the duffing oscillator having a soft characteristics this method is based on the computation of very accurate high order approximate periodic solutions of galerkin s type in conjunction with the corresponding stability analysis which is performed by computing the eigenvalues of a certain matrix related to the first variational equations the results obtained by this numerical method are in good agreement with those found previously by huberman and crutchfield who used an analog computer to obtain the phase portraits

noetherian properties of skew polynomial rings with binomial relations in this work we study standard finitely presented associative algebras over a fixed field k a restricted class of skew polynomial rings with quadratic relations considered in an earlier work of m artin and w schelter will be studied we call them binomial skew polynomial algebras we establish necessary and sufficient conditions for such an algebra to be a noetherian domain

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

period doubling solutions in the duffing oscillator a galerkin approach the period doubling solutions for the attractor introduced by huberman and crutchfield are studied in the relevant range for the forcing frequency by a numerical method based on the computation of galerkin fourier approximations with very high accuracy for answering in an affirmative manner the questions of the existence of an exact period doubled solution the error estimate and the transition from a stable to an unstable solution very precise solutions combined with periodic series representations are required it is shown that the algorithm which uses galerkin fourier approximations of high order is a very efficient one

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

regular subgroups of the affine group and asymmetric product of radical braces abstract in this paper we introduce the asymmetric product of radical braces a construction which extends the semidirect product of radical braces this new construction allows to obtain rather systematic constructions of regular subgroups of the affine group and in particular our approach allows to put in a more general context the regular subgroups constructed by heged s 2000 9

the unique labelling of states via mackey s theorem a new approach to finding unique labels for many electron states in multicentre systems is described and a particular example eight electrons in s states on eight centres is worked out in detail this approach is shown to provide a labelling with a more direct physical interpretation than has previously been found

unique labelling of many electron states in multicentre systems general methods of generating complete sets of states for l or 2n electrons in s states on n centres are described it is shown that a unique labelling of these states can always be obtained using a fictitious sn symmetry of the n centres explicit descriptions of eight seven and four electrons on eight centres are derived and the problem of determining a unique labelling for these states on rigid eight centre structures with cubic and lower symmetry is solved

homological properties of sklyanin algebras to a pair consisting of an elliptic curve and a point on it odeskii and feigin associate certain quadratic algebras sklyanin algebras having the hilbert series of a polynomial algebra in this paper we show that sklyanin algebras have good homological properties and we obtain some information about their so called linear modules we also show how the construction by odeskii and feigin may be generalized so as to yield other sklyanin type algebras

combinatorial solutions to the reflection equation abstract we use ring theoretic methods and methods from the theory of skew braces to produce set theoretic solutions to the reflection equation we also use set theoretic solutions to construct solutions of the parameter dependent reflection equation

classification and determination of irreducible bases for induced representations for chains of subgroups of finite groups the problem of classification and determination of standard irreducible bases for induced representations of an arbitrary finite group is discussed a general prescription for classification and determination of these bases for an arbitrary induced representation is proposed and an extra independent prescription for the special case of primitive representations is given relations between appropriate transformation coefficients are investigated and in particular a matrix similar to the racah recoupling matrix from theory of multiple coupling of angular momentum is introduced

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

counterexample to a conjecture about braces abstract we find an example of a finite solvable group in fact a finite p group in which is not possible to define a left brace structure or equivalently which is not an iyb group this answers a question posed by cedo jespers and del rio related to the yang baxter equation our argument is an improvement of an argument of rump using results about hopf galois extensions and lsa structures we explain explicitly the relation between these two areas of mathematics and brace theory hoping that it will be useful in the future

skew polynomial rings with binomial relations abstract in this paper we continue the study of a class of standard finitely presented quadratic algebrasaover a fixed fieldk called binomial skew polynomial rings we consider some combinatorial properties of the set of defining relationsfand their implications for the algebraic properties ofa we impose a condition called onfand prove that in this caseais a free module of finite rank over a strictly ordered noetherian domain we show that an analogue of the diamond lemma is true for one sided ideals of a skew polynomial ringawith condition we prove also that if the set of defining relationsfis square free then condition is necessary and sufficient for the existence of a finite groebner basis of every one sided ideal ina and for left and right noetherianness ofa as a corollary we find a class of finitely generated non commutative semigroups which are left and right noetherian

set theoretic solutions of the yang baxter equation rc calculus and garside germs abstract building on a result by w rump we show how to exploit the right cyclic law x y x z y x y z in order to investigate the structure groups and monoids attached with involutive nondegenerate set theoretic solutions of the yang baxter equation we develop a sort of right cyclic calculus and use it to obtain short proofs for the existence both of the garside structure and of the i structure of such groups we describe finite quotients that play for the considered groups the role that coxeter groups play for artin tits groups

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

a chebyshev polynomial method for optimal control with state constraints abstract this paper presents a numerical technique for solving non linear constrained optimal control problems the method extends previous contributions to non linear unconstrained optimal control problems and is based upon a chebyshev series expansion of state and control the differential and integral expressions from the system dynamics and the performance index the boundary conditions and other general conditions are converted into some algebraic equations state inequality constraints are transformed into equality constraints through the use of slack variables the technique may start from a feasible or non feasible trajectory and avoids problems of singular arcs the applicability is illustrated on two well known state variable inequality constrained optimal control problems extensions of the approach to problems with other equality and inequality constraints on state and control are described but have not yet been tested on practical examples

numerical solution of the controlled duffing oscillator by the pseudospectral method abstract this paper presents a numerical method for solving the controlled duffing oscillator the method can be extended to nonlinear calculus of variations and optimal control problems the method is based upon constructing the m th degree interpolation polynomials to approximate the state and the control vector using legendre gauss lobatto collocation points the differential and integral expressions which arise in the system dynamics the performance index and the boundary conditions are converted into some algebraic equations which can be solved for the unknown coefficients

symmetry coordinates for oh atomic shells the group factorisation and correlation theorem methods used previously to provide a unique labelling of atomic shell symmetry coordinates are shown to give also a simple means for deriving explicit expressions for these coordinates results are given for the 12 atom shell and the two types of 24 atom shells in oh symmetry

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

homology of left non degenerate set theoretic solutions to the yang baxter equation abstract this paper deals with left non degenerate set theoretic solutions to the yang baxter equation lnd solutions a vast class of algebraic structures encompassing groups racks and cycle sets to each such solution there is associated a shelf i e a self distributive structure which captures its major properties we consider two co homology theories for lnd solutions one of which was previously known in a reduced form for biracks only an explicit isomorphism between these theories is described for groups and racks we recover their classical co homology whereas for cycle sets we get new constructions for a certain type of lnd solutions including quandles and non degenerate cycle sets the co homologies split into the degenerate and the normalized parts we express 2 cocycles of our theories in terms of group cohomology and in the case of cycle sets establish connexions with extensions this leads to a construction of cycle sets with interesting properties

unique labelling of symmetry coordinates and cyclic regions in crystals earlier work on unique labelling schemes in oh symmetry is extended to include 48 atom shells and the symmetry coordinates of shells subject to periodicity conditions a new approach to labelling is introduced based on the concept of cyclic regions in a crystal these are described by finite factor groups of the crystallographic space group which contain the isogonal group as a subgroup some examples are worked out in detail

braces radical rings and the quantum yang baxter equation abstract non degenerate cycle sets are equivalent to non degenerate unitary set theoretical solutions of the quantum yang baxter equation we embed such cycle sets into generalized radical rings braces and study their interaction in this context we establish a galois theory between ideals of braces and quotient cycle sets our main result determines the relationship between two square free cycle sets operating transitively on each other