totally real minimal tori in in this paper we show that all totally real minimal tori in open image in new window correspond to doubly periodic finite gap solutions of the tzitzeica equation open image in new window using the results on the tzitzeica equation in integrable systems theory we describe explicitly these tori by prym theta functions

scattered data interpolation using c 2 supersplines of degree six we show how c2 supersplines of degree six can be used to interpolate hermite data at the vertices of a quadrangulation we also present error bounds that show that our method has full approximation order seven and compare its efficiency with other c2interpolation methods in the literature

a constant mean curvature surface and the dirac operator the shape of a surface with constant mean curvature cmc has been studied in mathematics and physics related to nonlinear integrable theory and harmonic map model theory in the study a fictitious linear dirac type operator appears as a tool of the calculus konopelchenko b g and taimanov i a 1996 j phys a math gen 29 1261 5 in this paper i confine the dirac field defined in to a thin surface embedded in and obtain a proper dirac operator for the thin surface then it completely agrees with the dirac type operator used in the calculus of the cmc surface theory in other words the mathematical dirac type operator is realized by a physical dirac particle

extending lazy 2 cocycles on hopf algebras and lifting projective representations afforded by them abstract we study some problems related to lazy 2 cocycles such as extension of lazy 2 cocycles to a drinfeld double and to a radford biproduct yetter drinfeld data obtained from lazy 2 cocycles lifting of projective representations afforded by lazy 2 cocycles

general twisting of algebras abstract we introduce the concept of pseudotwistor with particular cases called twistor and braided twistor for an algebra a u in a monoidal category as a morphism t a a a a satisfying a list of axioms ensuring that a t u is also an algebra in the category this concept provides a unifying framework for various deformed or twisted algebras from the literature such as twisted tensor products of algebras twisted bialgebras and algebras endowed with fedosov products pseudotwistors appear also in other topics from the literature e g durdevich s braided quantum groups and ribbon algebras we also focus on the effect of twistors on the universal first order differential calculus as well as on lifting twistors to braided twistors on the algebra of universal differential forms

a deformation of tori with constant mean curvature in to those in other space forms it is shown that tori with constant mean curvature in r 3 constructed by wente 7 can be deformed to tori with constant mean curvature in the hyperbolic 3 space or the 3 sphere

construction of quintic powell sabin spline quasi interpolants based on blossoming abstract by blossoming marsden s identity we investigate local quasi interpolation schemes for c 2 continuous quintic powell sabin splines represented with a normalized b spline basis as applications various families of discrete and differential quasi interpolants reproducing quintic polynomials are presented

two sided two cosided hopf modules and doi hopf modules for quasi hopf algebras let h be a finite dimensional quasi hopf algebra over a field k and a a right h comodule algebra we introduce the category of two sided hopf modules and prove that it is isomorphic to a module category we also show that two sided hopf modules are coalgebra over a certain comonad we introduce doi hopf modules and show that they are comodules over a certain coring if the underlying h module coalgebra is finite dimensional then doi hopf modules are modules over a certain smash products a similar result holds for two sided two cosided hopf modules

quasialgebra structure of the octonions abstract we show that the octonions are a twisting of the group algebra of z 2 z 2 z 2 in the quasitensor category of representations of a quasi hopf algebra associated to a group 3 cocycle in particular we show that they are quasialgebras associative up to a 3 cocycle isomorphism we show that one may make general constructions for quasialgebras exactly along the lines of the associative theory including quasilinear algebra representation theory and an automorphism quasi hopf algebra we study the algebraic properties of quasialgebras of the type which includes the octonions further examples include the higher 2n onion cayley algebras and examples associated to hadamard matrices

macro elements and stable local bases for splines on powell sabin triangulations macro elements of arbitrary smoothness are constructed on powell sabin triangle splits these elements are useful for solving boundary value problems and for interpolation of hermite data it is shown that they are optimal with respect to spline degree and we believe they are also optimal with respect to the number of degrees of freedom the construction provides local bases for certain superspline spaces defined over powell sabin refinements these bases are shown to be stable as a function of the smallest angle in the triangulation which in turn implies that the associated spline spaces have optimal order approximation power

lawson s genus two surface and meromorphic connections we investigate the lawson genus 2 surface by methods from integrable system theory to this surface we derive a globally defined meromorphic dpw potential which is determined up to two unknown functions depending only on the spectral parameter

action of truncated quantum groups on quasi quantum planes and a quasi associative differential geometry and calculus ifq is apth root of unity there exists a quasi coassociative truncated quantum group algebra whose indecomposable representations are the physical representations ofu q sl2 whose coproduct yields the truncated tensor product of physical representations ofu q sl2 and whoser matrix satisfies quasi yang baxter equations these truncated quantum group algebras are examples of weak quasitriangular quasi hopf algebras quasi quantum group algebras open image in new window we describe a space open image in new window of functions on the quasi quantum plane i e of polynomials in noncommuting complex coordinate functionsz a on which multiplication operatorsz a and the elements of open image in new window can act so thatz a will transform according to some representation f of open image in new window open image in new window can be made into a quasi associative graded algebra open image in new window on which elements of open image in new window act as generalized derivations in the special case of the truncatedu q sl2 algebra we show that the subspaces open image in new window of monomials inz a ofnth degree vanish forn p 1 and that open image in new window carries the 2j 1 dimensional irreducible representation of open image in new window ifn 2j j 0 1 2 1 2 p 2 assuming that the representation f of the quasi quantum group algebra gives rise to anr matrix with two eigenvalues we develop a quasi associative differential calculus on open image in new window this implies construction of an exterior differentiation a graded algebra open image in new window of forms and partial derivatives a quasi associative generalization of noncommutative differential geometry is introduced by defining a covariant exterior differentiation of forms it is covariant under open image in new window gauge transformations

hom l r smash products hom diagonal crossed products and the drinfeld double of a hom hopf algebra abstract we introduce the hom analogue of the l r smash product and use it to define the hom analogue of the diagonal crossed product when h is a finite dimensional hom hopf algebra with bijective antipode and bijective structure map we define the drinfeld double of h its algebra structure is a hom diagonal crossed product and it has all expected properties namely it is quasitriangular and modules over it coincide with left right yetter drinfeld modules over h

diagonal crossed products and l r smash products over weak hopf algebras the notions of the diagonal crossed product and the l r smash product over weak hopf algebras which can be seen as generalizations of smash products over weak hopf algebras are introduced and studied in particular the weak hopf algebraic structure of the drinfeld double of a finite dimensional weak hopf algebra which is a special case of diagonal crossed products is given moreover two canonical examples the kadison products and the connes moscovici products over weak hopf algebras are introduced and investigated

a hopf differential for constant mean curvature surfaces in s2 r and h2 r a basic tool in the theory of constant mean curvature cmc surfaces in space forms is the holomorphic quadratic differential discovered by h hopf in this paper we generalize this differential to immersed cmc surfaces in the product spaces s r and h r and prove a corresponding result about the geometry of immersed cmc spheres s in these target spaces

composite finite elements of class ck let be a triangulation of d r2 with vertices a ai 1 i n and pkn d v ck d t v t pn where pn polynomials of total degree n for u cr d r k and v finite dimensional subspace of ck d we consider the hermite problem hk a v find v v d v ai d u ai 1 i n k this problem is solvable in v pkn d iff n 4k 1 enisek when v pkn d 3 where 3 is a subtriangulation of obtained by subdividing each t into 3 triangles the problem has a solution for n 4k 1 e g the c1 cubic hct triangle and a p27 triangle with 31 parameters when v pkn d 6 where 6 is a subtriangulation of obtained by subdividing each t into 6 triangles the problem has a solution for n 3k 1 e g the c1 quadratic ps triangle and a p25 triangle with 31 parameters in both cases the construction needs the partial derivatives d u ai for 2k 1 the domain d is assumed to be polygonal compact and connected

gauge theory on nonassociative spaces we show how to do gauge theory on the octonions and other nonassociative algebras such as quasi r4 models proposed in string theory we use the theory of quasialgebras obtained by cochain twist introduced previously the gauge theory in this case is twisting equivalent to the usual gauge theory on the underlying classical space we give a general u 1 yang mills example for any quasialgebra and a full description of the moduli space of flat connections in this theory for the cube z23 and hence for the octonions we also obtain further results about the octonions themselves an explicit moyal product description of them as a nonassociative quantization of functions on the cube and a characterization of their cochain twist as invariant under fourier transform

on c 2 quintic spline functions over triangulations of powell sabin s type abstract given a triangulation of a polygonal domain we find a refinemet of of by choosing t in a neighborhood of the center of the inscribed circle of each triangle of t connecting t to the vertices of the triangle t and connecting t to t if t shares an interior edge with t or to the midpoint e of any boundary edge e of t the resulting triangulation is a triangulation of powell sabin s type we investigate a c 2 quintic spline space s 5 2 whose elements are c 3 only at t s we give a dimension formula for this spline space show how to construct a locally supported basis display an interpolation scheme and prove that this spline space has the full approximation order

frobenius and separable functors for the category of entwined modules over cowreaths ii applications abstract let h be a quasi hopf algebra we apply results obtained in 8 to give necessary and sufficient conditions for the forgetful functor from doi hopf modules two sided hopf modules or yetter drinfeld modules over h to representations of the underlying algebra to be frobenius resp separable we show that in some situations these conditions reduce to the unimodularity and or co semisimplicity of the quasi hopf algebra h

an explicit local basis for c 1 cubic spline spaces over a triangulated quadrangulation let s31 be the bivariate c1 cubic spline space over a triangulated quadrangulation in this paper an explicit representation of a locally supported basis of s31 is given using the interpolation conditions at vertices

quasi quantum groups knots three manifolds and topological field theory we show how to construct starting from a quasi hopf algebra or quasiquantum group invariants of knots and links in some cases these invariants give rise to invariants of the three manifolds obtained by surgery along these links this happens for a finite dimensional quasi quantum group whose definition involves a finite groupg and a 3 cocycle which was first studied by dijkgraaf pasquier and roche we treat this example in more detail and argue that in this case the invariants agree with the partition function of the topological field theory of dijkgraaf and witten depending on the same datag

generalized diagonal crossed products and smash products for quasi hopf algebras applications in this paper we introduce generalizations of diagonal crossed products two sided crossed products and two sided smash products for a quasi hopf algebra h the results we obtain may then be applied to h hopf bimodules and generalized yetter drinfeld modules the generality of our situation entails that the generating matrix formalism cannot be used forcing us to use a different approach this pays off because as an application we obtain an easy conceptual proof of an important but very technical result of hausser and nill concerning iterated two sided crossed products

from nurbs to nurps geometries abstract quadratic powell sabin splines and their rational extension the so called nurps surfaces are an interesting alternative for classical tensor product nurbs in the context of isogeometric analysis because they allow the use of local refinements while retaining a b spline like representation and exact description of conic sections in this paper we present a simple and effective strategy to convert a given planar geometry defined by a quadratic nurbs representation into a nurps representation suitable for the analysis

embedded constant mean curvature tori in the three sphere we prove that any constant mean curvature embedded torus in the three dimensional sphere is axially symmetric and use this to give a complete classification of such surfaces for any given value of the mean curvature

relative hopf modules for dual quasi hopf algebras univ bucharest fac math ro 70109 bucharest 1 romania limburgs univ ctr dept math b 3590 diepenbeek belgium bulacu d univ bucharest fac math str acad 14 ro 70109 bucharest 1 romania

stable hypersurfaces with constant scalar curvature it is well known that hypersurfaces m n with constant mean curvature in a riemannian manifold overline m n 1 c of constant sectional curvature c are solutions to the variational problem of extremizing the area function for volumepreserving variations

a bivariate c1 cubic super spline space on powell sabin triangulation a bivariate c 1 cubic super spline is constructed on powell sabin type 1 split with the additional smoothness at vertices in the original triangulation being c 2 which permits the hermite interpolation up to the second order partial derivatives exactly on all the vertices in the original triangulation the locally supported dual basis and computational details using derivatives around each vertex are given

braided cyclic cocycles and nonassociative geometry we use monoidal category methods to study the noncommutative geometry of nonassociative algebras obtained by a drinfeld type cochain twist these are the so called quasialgebras and include the octonions as braided commutative but nonassociative coordinate rings as well as quasialgebra versions cq g of the standard q deformation quantum groups we introduce the notion of ribbon algebras in the category which are algebras equipped with a suitable generalized automorphism and obtain the required generalization of cyclic cohomology we show that this braided cyclic cohomology is invariant under a cochain twist we also extend to our generalization the relation between cyclic cohomology and differential calculus on the ribbon quasialgebra the paper includes differential calculus and cyclic cocycles on the octonions as a finite nonassociative geometry as well as the algebraic noncommutative torus as an associative example

noncommutative finite dimensional manifolds i spherical manifolds and related examples we exhibit large classes of examples of noncommutative finite dimensional manifolds which are non formal deformations of classical manifolds the main result of this paper is a complete description of noncommutative three dimensional spherical manifolds a noncommutative version of the sphere s 3 defined by basic k theoretic equations we find a 3 parameter family of deformations of the standard 3 sphere s 3 and a corresponding 3 parameter deformation of the 4 dimensional euclidean space 4 for generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed 4 u only depend on two parameters and are isomorphic to the algebras introduced by sklyanin in connection with the yang baxter equation it follows that different can span the same this equivalence generates a foliation of the parameter space this foliation admits singular leaves reduced to a point these critical points are either isolated or fall in two 1 parameter families up to the simple operation of taking the fixed algebra by an involution these two families are identical and we concentrate here on c for the above isomorphism with the sklyanin algebra breaks down and the corresponding algebras are special cases of deformations a notion which we generalize in any dimension and various contexts and study in some detail here and this point is crucial the dimension is not an artifact i e the dimension of the classical model but is the hochschild dimension of the corresponding algebra which remains constant during the deformation besides the standard noncommutative tori examples of deformations include the recently defined noncommutative 4 sphere as well as m dimensional generalizations noncommutative versions of spaces and quantum groups which are deformations of various classical groups we develop general tools such as the twisting of the clifford algebras in order to exhibit the spherical property of the hermitian projections corresponding to the noncommutative dimensional spherical manifolds a key result is the differential self duality properties of these projections which generalize the self duality of the round instanton

hopf modules and the double of a quasi hopf algebra we give a different proof for a structure theorem of hausser and nill on hopf modules over quasi hopf algebras we extend the structure theorem to a classification of two sided two cosided hopf modules by yetter drinfeld modules which can be defined in two rather different manners for the quasi hopf case the category equivalence between hopf modules and yetter drinfeld modules leads to a new construction of the drinfeld double of a quasi hopf algebra as proposed by majid and constructed by hausser and nill

actions of monoidal categories and generalized hopf smash products let r be a k algebra and c a monoidal category assume given the structure of a c category on the category rm of left r modules that is the monoidal category c is assumed to act on the category rm by a coherently associative bifunctor c rm rm we assume that this bifunctor is right exact in its right argument in this setup we show that every algebra a respectively coalgebra c in c gives rise to an r ring a r respectively an r coring c r whose modules respectively comodules are the a modules respectively c comodules within the category rm we show that this very general scheme for constructing co associative co rings gives conceptual explanations for the double of a quasi hopf algebra as well as certain doubles of hopf algebras in braided categories each time avoiding ad hoc computations showing associativity

capillary surfaces in a convex cone we show that a compact embedded hypersurface s subset mathbb r n 1 with constant higher order mean curvature in a convex piecewise smooth cone c which is perpendicular to c is part of a hypersphere also we prove that an embedded disk type constant mean curvature surface s subset mathbb r 3 in a convex polyhedral cone c which makes constant contact angles with c is a spherical cap if c has at most five faces this condition on the number of faces can be dropped if c is a right cone over a regular n gon and the contact angles are the same on s

doubles of quasi quantum groups in dr1 drinfeld showed that any finite dimensional hopf algebra extends to a quasitriangular hopf algebra the quantum double of based on the construction of a so called diagonal crossed product developed by the authors in hn we generalize this result to the case of quasi hopf algebras as for ordinary hopf algebras as a vector space the quasi quantum double is isomorphic to where denotes the dual of we give explicit formulas for the product the coproduct the r matrix and the antipode on and prove that they fulfill drinfeld s axioms of a quasitriangular quasi hopf algebra in particular becomes an associative algebra containing as a quasi hopf subalgebra on the other hand is not a subalgebra of unless the coproduct on is strictly coassociative it is shown that the category of finite dimensional representations of coincides with what has been called the double category of modules by s majid m2 thus our construction gives a concrete realization of majid s abstract definition of quasi quantum doubles in terms of a tannaka krein like reconstruction procedure the whole construction is shown to generalize to weak quasi hopf algebras with now being linearly isomorphic to a subspace of

surfaces on lie groups on lie algebras and their integrability it is shown that the problem of the immersion of a 2 dimensional surface into a 3 dimensional euclidean space as well as then dimensional generalization of this problem is related to the problem of studying surfaces in lie groups and surfaces in lie algebras a particular case of the general formalism presented here implies that any surface can be characterized in terms of 2 2 matrices using an arbitrary parametrization it is also shown that this generality of parametrization is useful for studying integrable surfaces i e surfaces described by integrable equations in particular starting from a suitable lax pair i e a suitable integrable equation it is possible to construct explicitly large classes of integrable surfaces

l r smash product for quasi hopf algebras we introduce a more general version of the so called l r smash product and study its relations with other kinds of crossed products two sided smash and crossed product and diagonal crossed product we also give an interpretation of the l r smash product in terms of an l r twisting datum

stability of hypersurfaces of constant mean curvature in riemannian manifolds hypersurfaces m n with constant mean curvature in a riemannian manifold overline m n 1 display many similarities with minimal hypersurfaces of overline m n 1 they are both solutions to the variational problem of minimizing the area function for certain variations in the first case however the admissible variations are only those that leave a certain volume function fixed for precise definitions see sect 2 this isoperimetric character of the variational problem associated to hypersurfaces of constant mean curvature introduces additional complications in the treatment of stability of such hypersurfaces