measurable kac cohomology for bicrossed products we study the kac cohomology for matched pairs of locally compact groups this cohomology theory arises from the extension theory of locally compact quantum groups we prove a measurable version of the kac exact sequence and provide methods to compute the cohomology we give explicit calculations in several examples using results of moore and wigner

non semi regular quantum groups coming from number theory in this paper we study c algebraic quantum groups obtained through the bicrossed product construction examples using groups of adeles are given and they provide the first examples of locally compact quantum groups which are not semi regular the crossed product of the quantum group acting on itself by translations does not contain any compact operator we describe all corepresentations of these quantum groups and the associated universal c algebras on the way we provide several remarks on c algebraic properties of quantum groups and their actions

on locally compact quantum groups whose algebras are factors in this paper we are interested in examples of locally compact quantum groups m such that both von neumann algebras m and the dual m are factors there is a lot of known examples such that m m are respectively of type i1 i1 but there is no examples with factors of other types we construct new examples of type i1 ii1 ii1 ii1 and iii iii for each 2 0 1 also we show that there is no such example with m or m a nite factor a locally compact l c quantum group in the von neumann algebra setting see 6 7 is a pair m where m is a von neumann algebra and is a comultiplication on m with left and right invariant weight in a canonical way every l c group is a commutative l c quantum group and every von neu mann group algebra of a l c group is a cocommutative l c quantum group conversely every commutative or cocommutative l c quantum group is ob tained in this way our aim is to obtain new examples of l c quantum groups which are as far as possible from groups so we will be interested in the least commutative and cocommutative examples the formulation of the problem is the following given a pair of factors of a certain type x1 x2 is it possible to nd a l c quantum group m such that m is a type x1 factor and the dual algebra m is a type x2 factor in the sense of murray von neumann s and connes classication of factors there exists a lot of examples for the

the spatial rokhlin property for actions of compact quantum groups abstract we introduce the spatial rokhlin property for actions of coexact compact quantum groups on c algebras generalizing the rokhlin property for both actions of classical compact groups and finite quantum groups two key ingredients in our approach are the concept of sequentially split homomorphisms and the use of braided tensor products instead of ordinary tensor products we show that various structure results carry over from the classical theory to this more general setting in particular we show that a number of c algebraic properties relevant to the classification program pass from the underlying c algebra of a rokhlin action to both the crossed product and the fixed point algebra towards establishing a classification theory we show that rokhlin actions exhibit a rigidity property with respect to approximate unitary equivalence regarding duality theory we introduce the notion of spatial approximate representability for actions of discrete quantum groups the spatial rokhlin property for actions of a coexact compact quantum group is shown to be dual to spatial approximate representability for actions of its dual discrete quantum group and vice versa

curve and surface fitting models based on the diagonalizable differential systems abstract curve and surface fitting is an important problem in computer aided geometric design including many methods such as the b spline method the nurbs method and so on however many curves and surfaces in the natural or engineering fields need to be described by differential equations in this paper we propose a new curve and surface fitting method based on the homogeneous linear differential systems in order to approximate general curves or surfaces well the diagonalizable differential systems with variable coefficients are adopted which have explicit solutions the fitting algorithms are presented for curves and surfaces from discrete points some numerical examples show that the two algorithms can obtain good fitting accuracy as the b spline method

an interior point method for block angular optimization an interior point method for block angular optimization is developed and the convergence properties of the method are described a major motivation for such a method is that most of the computation is easily parallelized computational results are presented for a class of large scale linear programming models these models are multicommodity flow problems that arise from an air force military airlift command application and generate problems as large as 100 000 rows and 300 000 columns

families of non linear subdivision schemes for scattered data fitting and their non tensor product extensions abstract in this article families of non linear subdivision schemes are presented that are based on univariate polynomials up to degree three these families of schemes are constructed by using dynamic iterative re weighed least squares method these schemes are suitable for fitting noisy data with outliers the codes are designed in a python environment to numerically fit the given data points although these schemes are non interpolatory but have the ability to preserve the shape of the initial polygon in case of non noisy initial data the numerical examples illustrate that the schemes constructed by non linear polynomials give better performance than the schemes that are constructed by linear polynomials mustafa et al 2015 moreover the numerical examples show that these schemes have the ability to reproduce polynomials and do not cause over and under fitting of the data furthermore families of non linear bivariate subdivision schemes are also presented that are based on linear and non linear bivariate polynomials

on pentagon ten term and tetrahedron relations it is shown that the tetrahedron equation under the substitution where p 23 is the permutation operator is reduced to a pair of pentagon and one ten term equations on operators s and examples of infinite dimensional solutions are found o doubles of novikov which generalize the heisenberg double of a hopf algebra provide a particular algebraic solution to the problem

hopf von neumann algebra bicrossproducts kac algebra bicrossproducts and the classical yang baxter equations two groups g1 g2 are said to be a matched pair if each acts on the space of the other and these actions say obey a certain compatibility condition for every matched pair of locally compact groups g1 g2 we construct an associated coinvolutive hopf von neumann algebra m g1 l g2 by simultaneous cross product and cross coproduct for non trivial these bicrossproduct hopf von neumann algebras are non commutative and non cocommutative if the modules for the actions are also matched then these bicrossproducts are kac algebras in this case we show that the dual kac algebra is of the same form with the roles of g1 g2 and of interchanged examples exist with g1 a simply connected lie group and choices of g2 determined by suitable solutions of the classical yang baxter equations on the complexification of the lie algebra of g1

dynamic evaluation of free form curves and surfaces free form curves and surfaces represented by control points together with well defined basis functions are widely used in computer aided geometric design efficient evaluation of points and derivatives of free form curves and surfaces plays an important role in interactive rendering or cnc machining in this paper we show that free form curves with properly defined basis functions are the solutions of linear differential systems by employing typical numerical methods for solving the differential systems points and derivatives of free form curves and surfaces can be computed in a dynamical way compared with traditional methods for evaluating free form curves and surfaces there are two advantages of the proposed technique first the proposed method is universal and efficient for evaluating a large class of free form curves and surfaces second the evaluation needs only arithmetic operations even when the free form curves and surfaces are defined using some transcendental functions

from multiplicative unitaries to quantum groups ii abstract it is shown that all important features of a c algebraic quantum group a defined by a modular multiplicative w depend only on the pair a rather than the multiplicative unitary operator w the proof is based on thorough study of representations of quantum groups as an application we present a construction and study properties of the universal dual of a quantum group defined by a modular multiplicative unitary without assuming existence of haar weights

equivariant poincare duality for quantum group actions we extend the notion of poincare duality in kk theory to the setting of quantum group actions an important ingredient in our approach is the replacement of ordinary tensor products by braided tensor products along the way we discuss general properties of equivariant kk theory for locally compact quantum groups including the construction of exterior products as an example we prove that the standard podle sphere is equivariantly poincare dual to itself

a decomposition procedure based on approximate newton directions the efficient solution of large scale linear and nonlinear optimization problems may require exploiting any special structure in them in an efficient manner we describe and analyze some cases in which this special structure can be used with very little cost to obtain search directions from decomposed subproblems we also study how to correct these directions using decomposable preconditioned conjugate gradient methods to ensure local convergence in all cases the choice of appropriate preconditioners results in a natural manner from the structure in the problem finally we conduct computational experiments to compare the resulting procedures with direct methods

a cutting plane approach for large scale capacitated multi period facility location using a specialized interior point method we propose a cutting plane approach namely benders decomposition for a class of capacitated multi period facility location problems the novelty of this approach lies on the use of a specialized interior point method for solving the benders subproblems the primal block angular structure of the resulting linear optimization problems is exploited by the interior point method allowing the either exact or inexact efficient solution of large instances the consequences of different modeling conditions and problem specifications on the computational performance are also investigated both theoretically and empirically providing a deeper understanding of the significant factors influencing the overall efficiency of the cutting plane method the methodology proposed allowed the solution of instances of up to 200 potential locations one million customers and three periods resulting in mixed integer linear optimization problems of up to 600 binary and 600 millions of continuous variables those problems were solved by the specialized approach in less than one hour and a half outperforming other state of the art methods which exhausted the 144 gb of available memory in the largest instances

a new approach to induction and imprimitivity results given a closed quantum subgroup of a locally compact quantum group we study induction of unitary corepresentations of the quantum subgroup to the ambient quantum group more generally we study induction given a coaction of the quantum subgroup on a c algebra we prove imprimitivity theorems that unify the existing theorems for actions and coactions of groups this means that we deflne quantum homogeneous spaces as c algebras and that we prove morita equivalence of crossed products and homogeneous spaces we essentially use von neumann algebraic techniques to prove these morita equivalences between c algebras the theory of induced representations of locally compact l c groups was introduced by mackey 21 who discovered the imprimitivity theorem characterizing induced representations through the presence of a covariant representation of a homogeneous space riefiel 26 provided a modern approach using the language of hilbert c modules after the work of riefiel several induction procedures and imprimitivity results have been obtained both for actions and for coactions of groups we shall brie y review them below the purpose of this paper is to develop such an induction and imprimitivity machinery in the setting of locally compact quantum groups 16 17 in this way we provide a unifled approach to several results on actions and coactions of groups on c algebras at the same time our proofs in the setting of l c quantum groups are simpler than the classical proofs dealing with coactions of groups another motivation comes from quantum group theory meyer and nest have undertaken a reformulation of the baum connes conjecture 24 in which induction and restriction play a crucial role the development of induction and imprimitivity in the current paper should play an equally important role in the formulation of a baum connes conjecture for quantum groups

embeddable quantum homogeneous spaces abstract we discuss various notions generalizing the concept of a homogeneous space to the setting of locally compact quantum groups on the von neumann algebra level we recall an interesting duality for such objects studied earlier by m izumi r longo s popa for compact kac algebras and by m enock in the general case of locally compact quantum groups a definition of a quantum homogeneous space is proposed along the lines of the pioneering work of vaes on induction and imprimitivity for locally compact quantum groups the concept of an embeddable quantum homogeneous space is selected and discussed in detail as it seems to be the natural candidate for the quantum analog of classical homogeneous spaces among various examples we single out the quantum analog of the quotient of the cartesian product of a quantum group with itself by the diagonal subgroup analogs of quotients by compact subgroups as well as quantum analogs of trivial principal bundles the former turns out to be an interesting application of the duality mentioned above

tensor products of operator spaces abstract in this paper we lay the foundations for a systematic study of tensor products of subspaces of c algebras to accomplish this various notions of duality are introduced and employed elementary proofs of the complete injectivity of the haagerup norm and of the extension theorem for completely bounded maps are given pisier s gamma norms are examined and found to be special cases of the haagerup norm we identify the greatest operator space cross norm and show that the spatial tensor norm is the least operator space cross norm in an appropriate sense indeed most of the elementary theory of banach space tensor norms generalizes to the category of operator spaces

coordination in coarse grained decomposition the dominance of parallel computers in the supercomputer field provides great incentive for the development of parallel algorithms in mathematical programming fortunately parallel computers provide a good medium for the implementation of many types of decomposition algorithms for large scale optimization in this paper a variety of coordination methods are studied for coarse grained decomposition techniques some are variants of existing approaches that are designed to enhance rates of convergence and or parallel efficiency while others are novel methods that try to take maximum advantage of the parallel environment

extensions of hopf algebras and lie bialgebras let f g be finite dimensional lie algebras over a field of characteristic zero regard f and g the dual lie coalgebra of g as lie bialgebras with zero cobracket and zero bracket respectively suppose that a matched pair f g of lie bialgebras is given which has structure maps then it induces a matched pair u f ug of hopf algebras where u f is the universal envelope of f and ug is the hopf dual of ug we show that the group opext u f ug of cleft hopf algebra extensions associated with u f ug is naturally isomorphic to the group opext f g of lie bialgebra extensions associated with f g an exact sequence involving either of these groups is obtained which is a variation of the exact sequence due to g i kac if g g g there follows a bijection between the set ext u f ug of all cleft hopf algebra extensions of u f by ug and the set ext f g of all lie bialgebra extensions of f by g

iterative bundle based decomposition for large scale nonseparable convex optimization abstract there has been considerable research in solving large scale separable convex optimization problems in this paper we present an algorithm for large scale nonseparable smooth convex optimization problems with block angular linear constraints the solution of the problem is approximated by solving a sequence of structured separable quadratic programs the bundle based decomposition bbd method of robinson in prekopa a szelezsan j strazicky b eds system modelling and optimization springer 1986 pp 751 756 annals de institute henri poincare analyse non lineaire 6 1989 435 447 is applied to each separable quadratic program we implement the algorithm and present computational experience

a specialized interior point algorithm for multicommodity network flows despite the efficiency shown by interior point methods in large scale linear programming they usually perform poorly when applied to multicommodity flow problems the new specialized interior point algorithm presented here overcomes this drawback this specialization uses both a preconditioned conjugate gradient solver and a sparse cholesky factorization to solve a linear system of equations at each iteration of the algorithm the ad hoc preconditioner developed by exploiting the structure of the problem is instrumental in ensuring the efficiency of the method an implementation of the algorithm is compared to state of the art packages for multicommodity flows the computational experiments were carried out using an extensive set of test problems with sizes of up to 700 000 variables and 150 000 constraints the results show the effectiveness of the algorithm

scenarios and policy aggregation in optimization under uncertainty a common approach in coping with multiperiod optimization problems under uncertainty where statistical information is not really enough to support a stochastic programming model has been to set up and analyze a number of scenarios the aim then is to identify trends and essential features on which a robust decision policy can be based this paper develops for the first time a rigorous algorithmic procedure for determining such a policy in response to any weighting of the scenarios the scenarios are bundled at various levels to reflect the availability of information and iterative adjustments are made to the decision policy to adapt to this structure and remove the dependence on hindsight

extensions of locally compact quantum groups and the bicrossed product construction abstract in the framework of locally compact quantum groups we study cocycle actions we develop the cocycle bicrossed product construction starting from a matched pair of locally compact quantum groups we define exact sequences and establish a one to one correspondence between cocycle bicrossed products and cleft extensions in this way we obtain new examples of locally compact quantum groups

rigidity for von neumann algebras given by locally compact groups and their crossed products we prove the first rigidity and classification theorems for crossed product von neumann algebras given by actions of non discrete locally compact groups we prove that for arbitrary free probability measure preserving actions of connected simple lie groups of real rank one the crossed product has a unique cartan subalgebra up to unitary conjugacy we then deduce a w strong rigidity theorem for irreducible actions of products of such groups more generally our results hold for products of locally compact groups that are nonamenable weakly amenable and that belong to ozawa s class mathcal s

bundle based decomposition for large scale convex optimization error estimate and application to block angular linear programs robinson has proposed the bundle based decomposition algorithm to solve a class of structured large scale convex optimization problems in this method the original problem is transformed by dualization to an unconstrained nonsmooth concave optimization problem which is in turn solved by using a modified bundle method in this paper we give a posteriori error estimates on the approximate primal optimal solution and on the duality gap we describe implementation and present computational experience with a special case of this class of problems namely block angular linear programming problems we observe that the method is efficient in obtaining the approximate optimal solution and compares favorably with minos and an advanced implementation of the dantzig wolfe decomposition method

permuting sparse rectangular matrices into block diagonal form we investigate the problem of permuting a sparse rectangular matrix into block diagonal form block diagonal form of a matrix grants an inherent parallelism for solving the deriving problem as recently investigated in the context of mathematical programming lu factorization and qr factorization to represent the nonzero structure of a matrix we propose bipartite graph and hypergraph models that reduce the permutation problem to those of graph partitioning by vertex separator and hypergraph partitioning respectively our experiments on a wide range of matrices using the state of the art graph and hypergraph partitioning tools metis and patoh revealed that the proposed methods yield very effective solutions both in terms of solution quality and runtime

deformation of c algebras by cocycles on locally compact quantum groups given a c algebra a with a left action of a locally compact quantum group g on it and a unitary 2 cocycle on g we define a deformation a of a the construction behaves well under certain additional technical assumptions on the most important of which is regularity meaning that c0 g g is isomorphic to the algebra of compact operators on some hilbert space in particular then a is stably isomorphic to the iterated twisted crossed product g op g a also in good situations the c algebra a carries a left action of the deformed quantum group g and we have an isomorphism g a g a when g is a genuine locally compact group we show that the action of g on c0 g cr g is always integrable stronger assumptions of properness and saturation of the action imply regularity as an example we make a preliminary analysis of the cocycles on the duals of some solvable lie groups recently constructed by bieliavsky et al and discuss the relation of our construction to that of bieliavsky and gayral

quantum groups with projection on von neumann algebra level abstract we introduce an axiomatization of the notion of a semidirect product of locally compact quantum groups and study their properties our approach is slightly different from the one introduced in the thesis of s roy 16 and unlike the investigations of roy we work within the von neumann algebraic picture this allows the use of powerful techniques related to crossed products by actions of locally compact quantum groups in particular we show existence of a braided comultiplication on the algebra spanned by slices of braided multiplicative unitary

examples of non compact quantum group actions abstract we present two examples of actions of non regular locally compact quantum groups on their homogeneous spaces the homogeneous spaces are defined in a way specific to these examples but the definitions we use have the advantage of being expressed in purely c algebraic language we also discuss continuity of the obtained actions finally we describe in detail a general construction of quantum homogeneous spaces obtained as quotients by compact quantum subgroups