the dirichlet problem for nonlinear second order elliptic equations iii functions of the eigenvalues of the hessian on etudie le probleme de dirichlet dans un domaine borne de r n a frontiere lisse f d 2 u dans u sur

the mathematics of eigenvalue optimization optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge such problems arise often in theory and practice particularly in engineering design and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory this essay presents a personal choice of some central mathematical ideas outlined for the broad optimization community i discuss the convex analysis of spectral functions and invariant matrix norms touching briefly on semidefinite representability and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and lie algebra analogous nonconvex notions lead into eigenvalue perturbation theory the last third of the article concerns stability for polynomials matrices and associated dynamical systems ending with a section on robustness the powerful and elegant language of nonsmooth analysis appears throughout as a unifying narrative thread

simultaneous primal dual right hand side sensitivity analysis from a strictly complementary solution of a linear program this paper establishes theorems about the simultaneous variation of right hand sides and cost coefficients in a linear program from a strictly complementary solution some results are extensions of those that have been proven for varying the right hand side of the primal or the dual but not both other results are new in addition changes in the optimal partition and what that means in economic terms are related to the basis driven approach notably to the theory of compatibility in addition to new theorems about this relation the transition graph is extended to provide another visualization of the underlying economics

an augmented lagrangian interior point method using directions of negative curvature we describe an efficient implementation of an interior point algorithm for non convex problems that uses directions of negative curvature these directions should ensure convergence to second order kkt points and improve the computational efficiency of the procedure some relevant aspects of the implementation are the strategy to combine a direction of negative curvature and a modified newton direction and the conditions to ensure feasibility of the iterates with respect to the simple bounds the use of multivariate barrier and penalty parameters is also discussed as well as the update rules for these parameters we analyze the convergence of the procedure both the linesearch and the update rule for the barrier parameter behave appropriately as the main goal of the paper is the practical usage of negative curvature a set of numerical results on small test problems is presented based on these results the relevance of using directions of negative curvature is discussed

composite media and asymptotic dirichlet forms we introduce a metric topology on the space of dirichlet forms and study compactness and closure properties of families of local and non local forms

a note on the use of vector barrier parameters for interior point methods a key feature to ensure desirable convergence properties in an interior point method is the appropriate choice of an updating rule for the barrier parameter in this work we analyze and describe updating rules based on the use of a vector of barrier parameters we show that these updating rules are well defined and satisfy sufficient conditions to ensure convergence to the correct limit points we also present some numerical results that illustrate the improved performance of these strategies compared to the use of a scalar barrier parameter

the character table for a hopf algebra arising from the drinfel d double abstract in this paper we give a definition of the character table for a hopf algebra h over a commutative field k of characteristic 0 when h is a finite dimensional semisimple hopf algebra over an algebraically closed field k of characteristic 0 we obtain a formula for the character table which extends the notion of character table for a finite group

representations of finite dimensional hopf algebras abstract lethdenote a finite dimensional hopf algebra with antipodesover a field k we give a new proof of the fact due to os thathis a symmetric algebra if and only ifhis unimodular ands2is inner ifhis involutory and not semisimple then the dimensions of all projectiveh modules are shown to be divisible by char k in the case where k is a splitting field forh we give a formula for the rank of the cartan matrix ofh reduced mod char k in terms of an integral forh explicit computations of the cartan matrix the ring structure ofg0 h and the structure of the principal indecomposable modules are carried out for certain specific hopf algebras in particular for the restricted enveloping algebras of completely solvablep lie algebras and ofsl 2 k

on kaplansky s fifth conjecture abstract we prove that the antipode of a semisimple hopf algebra is an involution if the characteristic of the base field is very large

hyperbolic polynomials and interior point methods for convex programming hyperbolic polynomials have their origins in partial differential equations we show in this paper that they have applications in interior point methods for convex programming each homogeneous hyperbolic polynomial p has an associated open and convex cone called its hyperbolicity cone we give an explicit representation of this cone in terms of polynomial inequalities the function fx log px is a logarithmically homogeneous self concordant barrier function for the hyperbolicity cone with barrier parameter equal to the degree of p the function fx possesses striking additional properties that are useful in designing long step interior point methods for example we show that the long step primal potential reduction methods of nesterov and todd and the surface following methods of nesterov and nemirovskii extend to hyperbolic barrier functions we also show that there exists a hyperbolic barrier function on every homogeneous cone

superlinear convergence of primal dual interior point algorithms for nonlinear programming the local convergence properties of a class of primal dual interior point methods are analyzed these methods are designed to minimize a nonlinear nonconvex objective function subject to linear equality constraints and general inequalities they involve an inner iteration in which the log barrier merit function is approximately minimized subject to satisfying the linear equality constraints and an outer iteration that specifies both the decrease in the barrier parameter and the level of accuracy for the inner minimization under nondegeneracy assumptions it is shown that asymptotically for each value of the barrier parameter solving a single primal dual linear system is enough to produce an iterate that already matches the barrier subproblem accuracy requirements the asymptotic rate of convergence of the resulting algorithm is q superlinear and may be chosen arbitrarily close to quadratic furthermore this rate applies componentwise these results hold in particular for the method described in a r conn n i m gould d orban and p l toint math program ser b 87 2000 pp 215 249 and indicate that the details of its inner minimization are irrelevant in the asymptotics except for its accuracy requirements

determinantal representations of singular hypersurfaces in pn abstract a global determinantal representation of projective hypersurface x p n is a matrix whose entries are linear forms in homogeneous coordinates and whose determinant defines the hypersurface we study the properties of such representations for singular possibly reducible or non reduced hypersurfaces in particular we obtain the decomposability criteria for determinantal representations of globally reducible hypersurfaces further we classify the determinantal representations in terms of the corresponding kernel sheaves on x finally we extend the results to the case of symmetric self adjoint representations with implications to hyperbolic polynomials and the generalized lax conjecture

on the self concordance of the universal barrier function let k be a regular convex cone in hbox bbb r n and let f x be its universal barrier function let d kf x h ldots h be the k th order directional derivative at the point x in k 0 and direction h in hbox bbb r n we show that for every m ge3 there exists a constant c m 0 depending only on m such that d mf x h ldots h le c m d 2f x h h m 2 for m 3 this is the self concordance inequality of nesterov and nemirovskii our proof uses a powerful recent result of bourgain

a quadratically convergent polynomial algorithm for solving entropy optimization problems a potential reduction algorithm is developed for solving entropy optimization problems it is shown that the algorithm generates an epsilon optimal solution within at most o sqrt n log epsilon iterations where as usual n is the number of nonnegative variables and each iteration solves a system of linear equations under a computable criterion the algorithm is tuned to the pure newton method in a manner that leads to quadratic convergence while maintaining primal feasibility at each step a stopping criterion is derived which ensures that the objective function approaches its optimal value within any prescribed tolerance this applies for all entropy optimization problems having interior optimal solutions

multi parameter surfaces of analytic centers and long step surface following interior point methods we develop a long step polynomial time version of the method of analytic centers for non linear convex problems the method traces a multi parameter surface of analytic centers rather than the usual path which allows to handle cases with noncentered and possibly infeasible starting point

on the convergence of the central path in semidefinite optimization the central path in linear optimization always converges to the analytic center of the optimal set this result was extended to semidefinite optimization in d goldfarb and k scheinberg siam j optim 8 1998 pp 871 886 in this paper we show that this latter result is not correct in the absence of strict complementarity we provide a counterexample where the central path converges to a different optimal solution this unexpected result raises many questions we also give a short proof that the central path always converges in semidefinite optimization by using ideas from algebraic geometry

characterization of the barrier parameter of homogeneous convex cones we characterize the smallest best barrier parameter of self concordant barriers for homogeneous convex cones in particular we prove that this parameter is the same as the rank of the cone which is the number of steps in a recursive construction of the cone siegel domain construction we also provide lower bounds on the barrier parameter in terms of the caratheodory number of the cone the bounds are tight for homogeneous self dual cones 1998 the mathematical programming society inc published by elsevier science b v

self concordant barriers for hyperbolic means the geometric mean and the function det 1 m on the m by m positive definite matrices are examples of hyperbolic means functions of the form p 1 m where p is a hyperbolic polynomial of degree m a homogeneous polynomial p is hyperbolic with respect to a vector d if the polynomial t p x td has only real roots for every vector x any hyperbolic mean is positively homogeneous and concave on a suitable domain we present a self concordant barrier for its hypograph with barrier parameter o m2 our approach is direct and shows for example that the function mlog det 1 is an m2 self concordant barrier on a natural domain such barriers suggest novel interior point approaches to convex programs involving hyperbolic means

cone free primal dual path following and potential reduction polynomial time interior point methods we present a framework for designing and analyzing primal dual interior point methods for convex optimization we assume that a self concordant barrier for the convex domain of interest and the legendre transformation of the barrier are both available to us we directly apply the theory and techniques of interior point methods to the given good formulation of the problem as is without a conic reformulation using the very usual primal central path concept and a less usual version of a dual path concept we show that many of the advantages of the primal dual interior point techniques are available to us in this framework and therefore they are not intrinsically tied to the conic reformulation and the logarithmic homogeneity of the underlying barrier function

geometry of homogeneous convex cones duality mapping and optimal self concordant barriers we study homogeneous convex cones we first characterize the extreme rays of such cones in the context of their primal construction due to vinberg and also in the context of their dual construction due to rothaus then using these results we prove that every homogeneous cone is facially exposed we provide an alternative proof of a result of guler and tunc el that the siegel rank of a symmetric cone is equal to its caratheodory number our proof does not use the jordan von neumann wigner characterization of the symmetric cones but it easily follows from the primal construction of the homogeneous cones and our results on the geometry of homogeneous cones in primal and dual forms we study optimal self concordant barriers in this context we briefly discuss the duality mapping in the context of automorphisms of convex cones and prove using numerical integration that the duality mapping is not an involution on certain self dual cones

analytic centers and repelling inequalities the new concepts of repelling inequalities repelling paths and prime analytic centers are introduced a repelling path is a generalization of the analytic central path for linear programming and we show that this path has a unique limit furthermore this limit is the prime analytic center if the set of repelling inequalities contains only those constraints that shape the polytope because we allow lower dimensional polytopes the proof techniques are non standard and follow from data perturbation analysis this analysis overcomes the difficulty that analytic centers of lower dimensional polytopes are not necessarily continuous with respect to the polytope s data representation a second concept introduced here is that of the prime analytic center in which we establish its uniqueness in the absence of redundant inequalities again this is well known for full dimensional polytopes but it is not immediate for lower dimensional polytopes because there are many different data representations of the same polytope each without any redundant inequalities these two concepts combine when we introduce ways in which repelling inequalities can interact

simultaneous data perturbations and analytic center convergence the central path is an infinitely smooth parameterization of the nonnegative real line and its convergence properties have been investigated since the mid 1980s however the central path followed by an infeasible interior point method relies on three parameters instead of one and hence is a surface instead of a path the additional parameters are included to allow for simultaneous perturbations in the cost vectors and right hand side vectors this paper provides a detailed analysis of the perturbed central path that is followed by infeasible interior point methods and we characterize when such a path converges we develop a set hausdorff convergence property and show that the central paths impose an equivalence relation on the set of admissible cost vectors we conclude with a technique to test for convergence under arbitrary simultaneous data perturbations

barrier functions in interior point methods we show that the universal barrier function of a convex cone introduced by nesterov and nemirovskii is the logarithm of the characteristic function of the cone this interpretation demonstrates the invariance of the universal barrier under the automorphism group of the underlying cone this provides a simple method for calculating the universal barrier for homogeneous cones we identify some known barriers as the universal barrier scaled by an appropriate constant we also calculate some new universal barrier functions our results connect the field of interior point methods to several branches of mathematics such as lie groups jordan algebras siegel domains differential geometry complex analysis of several variables etc

on cone of nonsymmetric positive semidefinite matrices abstract in this paper we analyze and characterize the cone of nonsymmetric positive semidefinite matrices ns psd firstly we study basic properties of the geometry of the ns psd cone and show that it is a hyperbolic but not homogeneous cone secondly we prove that the ns psd cone is a maximal convex subcone of p 0 matrix cone which is not convex but the interior of the ns psd cone is not a maximal convex subcone of p matrix cone as the byproducts some new sufficient and necessary conditions for a nonsymmetric matrix to be positive semidefinite are given finally we present some properties of metric projection onto the ns psd cone

nonlocal effects in two dimensional conductivity the paper deals with the asymptotic behaviour as e 0 of a two dimensional conduction problem whose matrix valued conductivity ae is e periodic and not uniformly bounded with respect to e we prove that only under the assumptions of equi coerciveness and l1 boundedness of the sequence ae the limit problem is a conduction problem of same nature this new result points out a fundamental difference between the two dimensional conductivity and the three dimensional one indeed under the same assumptions of periodicity equi coerciveness and l1 boundedness it is known that the high conductivity regions can induce nonlocal effects in three or greater dimensions

sensitivity analysis in linear programming and semidefinite programming using interior point methods we analyze perturbations of the right hand side and the cost parameters in linear programming lp and semidefinite programming sdp we obtain tight bounds on the perturbations that allow interior point methods to recover feasible and near optimal solutions in a single interior point iteration for the unique nondegenerate solution case in lp we show that the bounds obtained using interior point methods compare nicely with the bounds arising from using the optimal basis we also present explicit bounds for sdp using the monteiro zhang family of search directions and specialize them to the aho h k m and nt directions

long step strategies in interior point primal dual methods in this paper we analyze from a unique point of view the behavior of path following and primal dual potential reduction methods on nonlinear conic problems we demonstrate that most interior point methods with o root n ln 1 epsilon efficiency estimate can be considered as different strategies of minimizing a convex primal dual potential function in an extended primal dual space their efficiency estimate is a direct consequence of large local norm of the gradient of the potential function along a central path it is shown that the neighborhood of this path is a region of the fastest decrease of the potential therefore the long step path following methods are in a sense the best potential reduction strategies we present three examples of such long step strategies we prove also an efficiency estimate for a pure primal dual potential reduction method which can be considered as an implementation of a penalty strategy based on a functional proximity measure using the convex primal dual potential we prove efficiency estimates for karmarkar type and dikin type methods as applied to a homogeneous reformulation of the initial primal dual problem

hermite s method of separation of solutions of systems of algebraic equations and its applications abstract the paper may be considered as a supplement to the book by m krein and m naimark problem find the number of real solutions of an algebraic system f1 x y 0 f2 x y 0 satisfying an algebraic condition g x y 0 we give a review of hermite s method which permits one to solve the problem using a finite number of algebraic operations on the coefficients of f1 f2 g the method reduces the problem to an investigation of the quadratic form in x0 xn 1 j 1 n j j x 0 x 1 j x n 1 j n 1 2 where j j is a solution of the system and n deg f1 deg f2 methods involved elimination theory the theory of symmetric functions and the theory of hankel quadratic forms applications analogue of sturm series in r 2 calculation of the kronecker poincare index of an algebraic field with respect to an algebraic curve in r 2 conditions for the sign definiteness of a homogeneous higher order polynomial of three variables we discuss also the possibility of extension of the method to r 2

homogenization of stiff plates and two dimensional high viscosity stokes equations the paper deals with the homogenization of stiff heterogeneous plates assuming that the coefficients are equi bounded in l 1 we prove that the limit of a sequence of plate equations remains a plate equation which involves a strongly local linear operator acting on the second gradients this compactness result is based on a div curl lemma for fourth order equations on the other hand using an intermediate stream function we deduce from the plates case a similar result for high viscosity stokes equations in dimension two so that the nature of the stokes equation is preserved in the homogenization process finally we show that the l 1 boundedness assumption cannot be relaxed indeed in the case of the stokes equation the concentration of one very rigid strip on a line induces the appearance of second gradient terms in the limit problem which violates the compactness result obtained under the l 1 boundedness condition

structured matrices continued fractions and root localization of polynomials we give a detailed account of various connections between several classes of objects hankel hurwitz toeplitz vandermonde and other structured matrices stietjes and jacobi type continued fractions cauchy indices moment problems total positivity and root localization of univariate polynomials along with a survey of many classical facts we provide a number of new results

determination of the closure of the set of elasticity functionals we determine the closure for mosco convergence in l2 3 of the set of elasticity functionals we prove that this closure coincides with the set of all non negative lower semicontinuous quadratic functionals which are objective i e which vanish for rigid motions the result is still valid if we consider only the set of isotropic elasticity functionals which have a prescribed poisson coefficient this shows that a very large family of materials can be reached when homogenizing a composite material with highly contrasted rigidity coefficients

primal dual symmetry and scale invariance of interior point algorithms for convex optimization we present a definition of symmetric primal dual algorithms for convex optimization problems expressed in the conic form after describing a generalization of the space approach for such optimization problems we show that a symmetric space approach can be developed for a convex optimization problem in the conic form if and only if the underlying cone is homogeneous and self dual we provide an alternative definition of self scaled barriers and then conclude with a discussion of the scalings of the variables which keep the underlying convex cone invariant

asymptotics of multivariate sequences part iii quadratic points abstract we consider a number of combinatorial problems in which rational generating functions may be obtained whose denominators have factors with certain singularities specifically there exist points near which one of the factors is asymptotic to a nondegenerate quadratic we compute the asymptotics of the coefficients of such a generating function the computation requires some topological deformations as well as fourier laplace transforms of generalized functions we apply the results of the theory to specific combinatorial problems such as aztec diamond tilings cube groves and multi set permutations

on the complexity of following the central path of linear programs by linear extrapolation ii a class of algorithms is proposed for solving linear programming problems withm inequality constraints by following the central path using linear extrapolation with a special adaptive choice of steplengths the latter is based on explicit results concerning the convergence behaviour of newton s method to compute points on the central pathx r r 0 and this allows to estimate the complexity i e the total numbern n r of steps needed to go from an initial pointx r to a final pointx r 0 by an integral of the local weighted curvature of the primal dual path here the central curve is parametrized with the logarithmic penalty parameterr 0 it is shown that for large classes of problems the complexity integral i e the number of stepsn is not greater than constm log r where 1 2 e g 1 4 or 3 8 note that 1 2 gives the complexity of zero order methods we also provide a lower bound for the complexity showing that for some problems the above estimation can hold only for 1 3 as a byproduct many analytical and structural properties of the primal dual central path are obtained there are for instance close relations between the weighted curvature and the logarithmic derivatives of the slack variables the dependence of these quantities on the parameterr is described also related results hold for a family of weighted trajectories into which the central path can be embedded

applications of the notion of analytic center in approximation estimation problems abstract a new method for solving a set of linear inequalities based on a generalization of the central maximum entropy solution of moment problems is applied to problems of numerical approximation cubature and filtering

smooth hyperbolicity cones are spectrahedral shadows hyperbolicity cones are convex algebraic cones arising from hyperbolic polynomials a well understood subclass of hyperbolicity cones is that of spectrahedral cones and it is conjectured that every hyperbolicity cone is spectrahedral in this paper we prove a weaker version of this conjecture by showing that every smooth hyperbolicity cone is the linear projection of a spectrahedral cone that is a spectrahedral shadow

an interior point approach to sensitivity analysis in degenerate linear programs we consider an interior point approach to sensitivity analysis in linear programming developed by the authors we investigate the quality of the interior point bounds under degeneracy in the case of a special type of degeneracy we show that these bounds have the same nice asymptotic relationship with the optimal partition bounds as in the nondegenerate case we prove a weaker relationship for general degenerate linear programs

an efficient algorithm for minimizing a sum of p norms we study the problem of minimizing a sum of p norms where p is a fixed real number in the interval 1 infty several practical algorithms have been proposed to solve this problem however none of them has a known polynomial time complexity in this paper we transform the problem into standard conic form unlike those in most convex optimization problems the cone for the p norm problem is not self dual unless p 2 nevertheless we are able to construct two logarithmically homogeneous self concordant barrier functions for this problem the barrier parameter of the first barrier function does not depend on p the barrier parameter of the second barrier function increases with p using both barrier functions we present a primal dual potential reduction algorithm to compute an epsilon optimal solution in polynomial time that is independent of p computational experiences with a matlab implementation are also reported

on the dimension of the set of rim perturbations for optimal partition invariance two new dimension results are presented for linear programs it is shown that the sum of the dimension of the optimal set and the dimension of the set of objective perturbations for which the optimal partition is invariant equals the number of variables a decoupling principle shows that the primal and dual results are additive the main result is then extended to convex quadratic programs but the dimension relationships are no longer dependent only on problem size furthermore although the decoupling principle does not extend completely the dimensions are additive as in the linear case

combining search directions using gradient flows the efficient combination of directions is a significant problem in line search methods that either use negative curvature or wish to include additional information such as the gradient or different approximations to the newton direction

an interior point perspective on sensitivity analysis in semidefinite programming we study the asymptotic behavior of the interior point bounds arising from the work of yildirim and todd on sensitivity analysis in semidefinite programming in comparison with the optimal partition bounds we introduce a weaker notion of nondegeneracy and discuss its implications for perturbations of the right hand side vector or the cost matrix we show that the interior point bounds evaluated on the central path using the monteiro zhang family of search directions converge as the duality gap tends to zero to the symmetrized version of the optimal partition bounds under mild nondegeneracy assumptions furthermore our analysis does not assume strict complementarity as long as the central path converges to the analytic center in a relatively controlled manner we also show that the same convergence results carry over to iterates lying in an appropriate very narrow central path neighborhood if the nesterov todd direction is used to evaluate the interior point bounds we extend our results to the case of simultaneous perturbations of the right hand side vector and the cost matrix we also provide examples illustrating that our assumptions in general cannot be weakened

invariance and efficiency of convex representations we consider two notions for the representations of convex cones g representation and lifted g representation the former represents a convex cone as a slice of another the latter allows in addition the usage of auxiliary variables in the representation we first study the basic properties of these representations we show that some basic properties of convex cones are invariant under one notion of representation but not the other in particular we prove that lifted g representation is closed under duality when the representing cone is self dual we also prove that strict complementarity of a convex optimization problem in conic form is preserved under g representations then we move to study efficiency measures for representations we evaluate the representations of homogeneous convex cones based on the smoothness of the transformations mapping the central path of the representation to the central path of the represented optimization problem

finite dimensional cosemisimple hopf algebras in characteristic 0 are semisimple finite dimensional cosemisimple hopf algebras are a natural generalization of group algebras much of the representation theory of finite groups can be extended to such hopf algebras however there are still many unanswered basic questions for such hopf algebras one such question is a conjecture mentioned by kaplansky i the square of the antipode of a finite dimensional cosemisimple hopf algebra is the identity in this paper we prove that the fourth power of the antipode is the identity in a finite dimensional cosemisimple hopf algebra over a field of characteristic 0 in the course of proving this we investigate two elements of the hopf algebra which are roughly analogous to the sum of the group elements in a group algebra we show that kaplansky s conjecture is equivalent to these two elements being equal for a finite dimensional hopf algebra a with antipode s over a field let l p q pq for p q e a denote the left module action of a on itself in this paper we study two elements of a defined by the trace function the element i defined by p a tr l p s for all pe a and the element x defined by p x tr l p for all p e a we show that 2 is a left integral and use it to prove theorem 3 3 the main result of this paper a finite dimensional cosemisimple hopf algebra over a field of characteristic 0 is also semisimple hence its antipode has order 1 2 or 4 we extend the first corollary to 3 proposition 91 in theorems 4 3 and 4 4 by finding necessary and sufficient conditions for s to be a non zero left integral in which case x a and also study how x relates to the structure of a in particular when dim a 1 0 the right ideal xa may be thought of as a measure of the extent to which s i the elements ji and k are two examples of elements of a associated with an endomorphism of a in general for f end a we define a e a by p a tr l p cf for all pea thus i as2 and x a in section 2 267 0021 8693 88 33 00

on self concordant barrier functions for conic hulls and fractional programming given a self concordant barrier function for a convex set figure not available see fulltext we determine a self concordant barrier function for the conic hull figure not available see fulltext of figure not available see fulltext as our main result we derive an optimal barrier for figure not available see fulltext based on the barrier function for figure not available see fulltext important applications of this result include the conic reformulation of a convex problem and the solution of fractional programs by interior point methods the problem of minimizing a convex concave fraction over some convex set can be solved by applying an interior point method directly to the original nonconvex problem or by applying an interior point method to an equivalent convex reformulation of the original problem our main result allows to analyze the second approach showing that the rate of convergence is of the same order in both cases

physics for algebraists non commutative and non cocommutative hopf algebras by a bicrossproduct construction abstract the initial part of this paper presents physics for algebraists in the context of quantum mechanics combined with gravity such physical notions as the yang baxter equations position observables momentum space momentum and position quantization etc are described many readers may wish to just read this initial part of the paper the physics leads to the search for self dual algebraic structures and finally to non commutative and non cocommutative hopf algebras by a bicrossproduct construction the entire paper contains numerous examples the non commutative and non cocommutative hopf algebras are obtained as a simultaneous smash product and smash coproduct and denoted h1 h2 among the examples is one obtained by modifying the weyl algebra we also give the context in which the compatibility requirements on the structure maps reduce to the classical yang baxter equations and an example related to drinfel d s double hopf algebra d h