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

cyclic hopf orders defined by isogenies of formal groups using degree 2 polynomial formal groups we construct hopf algebras over valuation rings of local number fields k that are orders in kg g cyclic of order p n the construction does not yield all greither orders when n 2 but does yield new classes of hopf orders for all n 3 and shows that a general hopf order in kg g cyclic of order p n involves at least n n 1 2 parameters

cleft extensions of hopf algebras ii let k be a p adic field let g and u be groups of order p and let d respectively m be a hopf order in the group algebra kg respectively in the algebra of maps u k we use the algebraic machinery of 2 to determine the cleft hopf algebra extensions of m by d and investigate which of these cleft extensions are hopf orders in a group algebra

induced hopf galois structures for a nite galois extension k k and an intermediate eld f such that gal k f has a normal complement in gal k k we construct and characterize hopf galois structures on k k which are induced by a pair of hopf galois structures on k f and f k

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

counting hopf galois structures on cyclic field extensions of squarefree degree abstract we investigate hopf galois structures on a cyclic field extension l k of squarefree degree n by a result of greither and pareigis each such hopf galois structure corresponds to a group of order n whose isomorphism class we call the type of the hopf galois structure we show that every group of order n can occur and we determine the number of hopf galois structures of each type we then express the total number of hopf galois structures on l k as a sum over factorisations of n into three parts as examples we give closed expressions for the number of hopf galois structures on a cyclic extension whose degree is a product of three distinct primes there are several cases depending on congruence conditions between the primes we also consider one case where the degree is a product of four primes

hopf galois structures on extensions of degree p2q and skew braces of order p2q the cyclic sylow p subgroup case abstract let p q be distinct primes with p 2 we classify the hopf galois structures on galois extensions of degree p 2 q such that the sylow p subgroups of the galois group are cyclic this we do according to greither and pareigis and byott by classifying the regular subgroups of the holomorphs of the groups g of order p 2 q in the case when the sylow p subgroups of g are cyclic this is equivalent to classifying the skew braces g furthermore we prove that if g and are groups of order p 2 q with non isomorphic sylow p subgroups then there are no regular subgroups of the holomorph of g which are isomorphic to equivalently a galois extension with galois group has no hopf galois structures of type g our method relies on the alternate brace operation on g which we use mainly indirectly that is in terms of the functions g aut g defined by g x x g g 1 these functions are in one to one correspondence with the regular subgroups of the holomorph of g and are characterised by the functional equation g h h g h for g h g we develop methods to deal with these functions with the aim of making their enumeration easier and more conceptual

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

cayley s theorem and hopf galois structures for semidirect products of cyclic groups abstract for g any finite group the left and right regular representations respectively of g into perm g map g into inhol g g inn g we determine regular embeddings of g into inhol g modulo equivalence by conjugation in hol g by automorphisms of g for groups g that are semidirect products g z h z k of cyclic groups and have trivial centers if h is the power of an odd prime p then the number of equivalence classes of regular embeddings of g into inhol g is equal to twice the number of fixed point free endomorphisms of g and we determine that number each equivalence class of regular embeddings determines a hopf galois structure on a galois extension of fields l k with galois group g we show that if h 1 is the hopf algebra that gives the standard non classical hopf galois structure on l k then h 1 gives a different hopf galois structure on l k for each fixed point free endomorphism of g

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

hopf galois structures on separable field extensions of odd prime power degree abstract we study hopf galois structures on separable field extensions l k of degree p n p an odd prime and denote by g the galois group of the normal closure over k we prove that if l k has a hopf galois structure of cyclic type then it has no structure of noncyclic type for n 3 we characterize the transitive groups g such that l k has cyclic hopf galois structures we prove that if l k has a nonabelian hopf galois structure of type n then it has an abelian structure whose type has the same exponent as n we obtain that for p 3 the two abelian noncyclic hopf galois structures do not occur on the same extension finally we list all possible sets of hopf galois structure types on a separable extension of degree p 3 for p 3 a prime

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

regularity of free boundaries in anisotropic capillarity problems and the validity of young s law local volume constrained minimizers in anisotropic capillarity problems develop free boundaries on the walls of their containers we prove the regularity of the free boundary outside a closed negligible set showing in particular the validity of young s law at almost every point of the free boundary our regularity results are not specific to capillarity problems and actually apply to sets of finite perimeter and thus to codimension one integer rectifiable currents arising as minimizers in other variational problems with free boundaries

hopf galois structures on field extensions of degree twice an odd prime square and their associated skew left braces abstract we determine the hopf galois structures on a galois field extension of degree twice an odd prime square and classify the corresponding skew left braces besides we determine the separable field extensions of degree twice an odd prime square allowing a cyclic hopf galois structure and the number of these structures

elementary abelian hopf galois structures and polynomial formal groups abstract we find a relationship between regular embeddings of g an elementary abelian p group of order p n into unipotent upper triangular matrices with entries in f p and commutative dimension n degree 2 polynomial formal groups with nilpotent upper triangular structure matrices we classify the latter when n 3 up to linear isomorphism and use that classification to determine the number of hopf galois structures on a galois extension l k of fields with galois group g elementary abelian of order p 3 we also obtain a lower bound on the number of hopf galois structures on a galois extension l k when the galois group is elementary abelian of order p n

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

minimizers of anisotropic surface tensions under gravity higher dimensions via symmetrization we consider a variational model describing the shape of liquid drops and crystals under the influence of gravity resting on a horizontal surface making use of anisotropic symmetrization techniques we establish the existence convexity and symmetry of minimizers for a class of surface tensions admissible to the symmetrization procedure in the case of smooth surface tensions we obtain the uniqueness of minimizers via an ode characterization

the perimeter inequality under steiner symmetrization cases of equality steiner symmetrization is known not to increase perimeter of sets in rn the sets whose perimeter is preserved under this symmetrization are characterized in the present paper

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

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

skew braces and the galois correspondence for hopf galois structures abstract let l k be a galois extension of fields with galois group and suppose l k is also an h hopf galois extension using the recently uncovered connection between hopf galois structures and skew left braces we introduce a method to quantify the failure of surjectivity of the galois correspondence from subhopf algebras of h to intermediate subfields of l k given by the fundamental theorem of hopf galois theory suppose l k h l n where n g then there exists a skew left brace g where g we show that there is a bijective correspondence between the set of intermediate fields e between k and l that correspond to k subhopf algebras of h and a set of sub skew left braces of g that we call the stable subgroups of g counting these subgroups and comparing that number with the number of subgroups of g describes how far the galois correspondence for the h hopf galois structure is from being surjective the method is illustrated by a variety of examples

equilibrium shapes for planar crystals in an external field the equilibrium shape of a two dimensional crystal in a convex background potential g x is analyzed for g 0 the shape of minimum energy may be deduced from surface tension via the wulff construction but if g is not constant little is known beyond the case of a crystal sitting in a uniform field only an unpublished result of okikiolu shows each connected component of the equilibrium crystal to be convex here it will be shown that any such component minimizes energy uniquely among convex sets of its area if the wulff shape and g x are symmetric under x x it follows that the equilibrium crystal is unique convex and connected this last result leads to a new proof that convex crystals away from equilibrium remain convex as they evolve by curvature driven flow subsequent work with felix otto shows without assuming symmetry that no equilibrium crystal has more than two convex components

the wulff theorem revisited the parametrized indicator measures and the brunn minkowski inequality are used to prove that the wulff set w is a minimizer for the surface energy where the density is the support function of w the support of the indicator measures associated to minimizing sequences is characterized it is shown that if w is polyhedral then minimizing sequences cannot oscillate

hopf galois structures of squarefree degree abstract let n be a squarefree natural number and let g be two groups of order n we determine the number of hopf galois structures of type g admitted by a galois extension of fields with galois group isomorphic to we give some examples including a full treatment of the case where n is the product of three primes

an exponential heat balance integral method implementations of the heat balance integral method are discussed in which exponential functions are used in place of the familiar polynomial approximants the rationale is based upon that of least squares in that the use of appropriate basis functions can enhance solution accuracy whilst this is true in principle it is shown that considerable skill must be exercised when deviating from polynomial approximants the discussions are illustrated by application to a familiar single phase stefan problem that is typical of heat transfer problems exhibiting decay like spatial solution profiles

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

monogenic hopf orders and associated orders of valuation rings abstract let r be a complete discrete valuation ring of mixed characteristic with perfect residue field and let h be a finite local commutative r hopf algebra we consider when there exists a finite extension of the field of fractions of r whose valuation ring is a galois h object if this occurs then h is monogenic conversely if h is also cocommutative and h is monogenic then there exists a valuation ring which is a galois h object to prove this result we represent h as the kernel of an isogeny of a special type between formal groups over r we deduce that if a is a finite abelian r hopf algebra such that both a and its dual are local then a is the associated order of a valuation ring if and only if the dual of a is monogenic

behavior of the solution of a stefan problem by changing thermal coefficients of the substance abstract we consider a one dimensional one phase stefan problem for a semi infinite substance we suppose that there is a transient heat flux at the fixed face and the thermal coefficients are constant the goal of this paper is to determine the behavior of the free boundary and the temperature by changing the thermal coefficients we use the maximum principle in order to obtain properties of monotony with respect to the latent heat of fusion the specific heat and the mass density we compute approximate solutions through the quasi stationary the goodman s heat balance integral and the biot s variational methods and a numerical solution through a finite difference scheme we show that the solution is not monotone with respect to the thermal conductivity the results obtained are important in technological applications

hopf galois structures on almost cyclic field extensions of 2 power degree let l k be a finite separable field extension and let e be the normal closure of l k let g gal e k and g gal e l we call l k almost cyclic if g has a normal cyclic complement in g this includes the case that l k is a cyclic galois extension or a radical extension we give a method for counting hopf galois structures on an almost cyclic extension l k we then count the hopf galois structures on an almost cyclic extension of degree 2n n 3 and determine how many of them are almost classical this is analogous to a result of t kohl t kohl classification of the hopf galois structures on prime power radical extensions j algebra 207 1998 525 546 which counts the hopf galois structures on a radical extension of odd prime power degree in contrast to the odd prime power degree case however we find that an almost cyclic extension l k of 2 power degree has hopf galois structures for which the hopf algebra acting on l is not commutative

a uniqueness proof for the wulff theorem the wulff problem is a generalisation of the isoperimetric problem and is relevant for the equilibrium of small elastic crystals it consists in minimising the generally anisotropic surface energy among sets of given volume a solution of this problem is given by a geometric construction due to wulff in the class of sets of finite perimeter this was first shown by j e taylor who using methods of geometric measure theory also proved uniqueness here a more analytic uniqueness proof is presented the main ingredient is a sharpened version of the brunn minkowski inequality

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

counting hopf galois structures on non abelian galois field extensions let l be a field which is a galois extension of the field k with galois group g greither and pareigis gp87 showed that for many g there exist k hopf algebras h other than the group ring kg which make l into an h hopf galois extension of k or a galois h object in the sense of chase and sweedler cs69 using galois descent they translated the problem of determining the hopf galois structures on l k into one which depends only on the galois group g by this translation they showed for example that any galois extension with non abelian g admits at least one non classical hopf galois structure byott by96 further translated the problem to a more amenable group theoretic problem and showed that a galois extension l k of fields with group g has a unique hopf galois structure namely that by kg iff n the order of g is a burnside number that is is coprime to n euler s phi function of n this implies that g is cyclic of square free order

hybrids of the heat balance integral method abstract a framework for refining and hybridizing the heat balance integral method is proposed while showing the non uniqueness of a combined integral method in the sense of myers and mitchell t g myers s l mitchell application of the combined integral method to stefan problems applied mathematical modelling 35 9 2011 4281 4294 it is evinced through the hybrids advanced and benchmarks undertaken that for the class of finite puiseux series commonly employed there are no globally efficient exponents synthesis of local regions of high accuracy of these hybrids is realized through the introduction of an applicable splicing algorithm

equilibrium shapes of crystals in a gravitational field crystals on a table we consider the variational problem associated with the equilibrium shape of crystals resting on a table in a gravitational field for two dimensional crystals the shape can be calculated explicitly i e reduced to quadrature in three dimensions only qualitative results are available the most interesting new result is that for large crystals under suitable conditions the top may be a corrugated facet or curved surface the motivation for this work comes from lowtemperature experiments on helium crystals in equilibrium with the superfluid

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

integral hopf galois structures on degree p2 extensions of p adic fields abstract let l k be a totally ramified normal extension of p adic fields of degree p2 we investigate the behavior of the valuation ring o l in the various hopf galois structures on l k specifically we determine when o l is hopf galois with respect to a hopf order in the corresponding hopf algebra when this occurs o l is necessarily a free module over this hopf order we also determine which hopf orders can arise in this way for cyclic extensions l k of degree p2 l n childs has shown under certain restrictions on the ramification numbers that if o l is hopf galois with respect to a hopf order in one of the hopf galois structures on l k then the same is true in all p hopf galois structures on l k we show that this no longer holds if the ramification conditions are relaxed or if elementary abelian extensions of degree of p2 are considered we illustrate our results with a special family of kummer extensions and with certain extensions arising from lubin tate formal groups

classification of the hopf galois structures on prime power radical extensions abstract let p be an odd prime and n a positive integer and let k be a field of characteristic zero let k k w with w p n a k where a is such that k k p n and let r denote the largest integer between 0 and n such that k k p r k p r where p r denotes a primitive p r th root of unity the extension k k is separable but not necessarily normal and by greither and pareigis is h galois with h a k hopf algebra form of a group ring kn where k is the normal closure of k k h is said to be almost classical if n k k the result is that if r n then there are p r hopf galois structures on k k for which the associated group n is cyclic of order p n of these p min r n r are almost classical and the rest are non almost classical when r n there are p n 1 h galois structures for which n c p n of which only one is almost classical finally we show that these are the only structures possible that is for this class of extensions n must be cyclic

heat balance integral methods applied to the one phase stefan problem with a convective boundary condition at the fixed face abstract in this paper we consider a one dimensional one phase stefan problem corresponding to the solidification process of a semi infinite material with a convective boundary condition at the fixed face the exact solution of this problem available recently in the literature enable us to test the accuracy of the approximate solutions obtained by applying the classical technique of the heat balance integral method and the refined integral method assuming a quadratic temperature profile in space we develop variations of these methods which turn out to be optimal in some cases throughout this paper a dimensionless analysis is carried out by using the parameters stefan number ste and the generalized biot number bi in addition it is studied the case when bi goes to infinity recovering the approximate solutions when a dirichlet condition is imposed at the fixed face some numerical simulations are provided in order to estimate the errors committed by each approach for the corresponding free boundary and temperature profiles

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

application of standard and refined heat balance integral methods to one dimensional stefan problems the work in this paper concerns the study of conventional and refined heat balance integral methods for a number of phase change problems these include standard test problems both with one and two phase changes which have exact solutions that enable us to test the accuracy of the approximate solutions we also consider situations where no analytical solution is available and compare these to numerical solutions it is popular to use a quadratic profile as an approximation of the temperature but we show that a cubic profile seldom considered in the literature is far more accurate in most circumstances in addition the refined integral method can give greater improvement still and we develop a variation on this method which turns out to be optimal in some cases we assess which integral method is better for various problems showing that it is largely dependent on the specified boundary conditions

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

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

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

duality for hopf orders in this paper we use duality to construct new classes of hopf orders in the group algebra kc p3 where k is a finite extension of q p and c p3 denotes the cyclic group of order p 3 included in this collection is a subcollection of hopf orders which are realizable as galois groups

nilpotent and abelian hopf galois structures on field extensions abstract let l k be a finite galois extension of fields with group when is nilpotent we show that the problem of enumerating all nilpotent hopf galois structures on l k can be reduced to the corresponding problem for the sylow subgroups of we use this to enumerate all nilpotent resp abelian hopf galois structures on a cyclic extension of arbitrary finite degree when is abelian we give conditions under which every abelian hopf galois structure on l k has type we also give a criterion on n such that every hopf galois structure on a cyclic extension of degree n has cyclic type

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

numerical schemes for one dimensional stefan like problems with a forcing term variable space grid and boundary immobilisation schemes based on the explicit finite difference method are applied to the one phase stefan like problems with a forcing term in order to evaluate the temperature distribution and the interface movement location and speed the numerical results obtained by the two schemes have been compared with the exact values and are found to be in good agreement with each other it is shown that the numerical solutions exhibit the expected convergence to the exact one as the mesh size is reduced a von neumann stability analysis of each scheme is also investigated

on frohlich s conjecture for rings of integers of tame extensions where is the complex conjugate of z the constant w z is called the artin root number of in m l j martinet showed that om is not always a free zf module when n q and when f is a quaternion group of order 8 subsequently in f5 a frtihlich showed that the class of u in this special case is determined by the sign of the artin root number of the irreducible sympletic character of 2 we now return to the general case and let ci tif be the class group of locally free zf modules given a locally free 2 f module x we denote its class by x philippe cassou nogu6s defined a class t w in ci f which is defined solely in terms of the values of artin root numbers of sympletic characters this class will be defined in w after numerous computations by fr6hlich et al cf f6 c t3 for instance fr6hlich conjectured the result given below which we prove in this paper

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