induced i l fuzzy topological spaces abstract in this paper we introduce the notion of induced i l fuzzy topological spaces by using the i l valued lower semicontinuous mappings and investigate some of their basic properties in particular we give a necessary and sufficient condition for an i l fuzzy topological space to be induced and prove that this kind of induced space preserves the cartesian product and the n compactness as well

stable bundles and integrable systems on considere la geometrie symplectique des fibres cotangents aux espaces de modules de fibres vectoriels stables sur une surface de riemann on montre que ce sont des systemes dynamiques hamiltoniens algebriquement completement integrables

stable subconstructs of fts iii exotic cases abstract in this paper we study stable subconstructs of fts which neither contain nor are contained in top hence in contrast to similar subconstructs studied previously they do not constitute a viable alternative theory of fuzzy topology for that reason we call them exotic

the g generalized hitchin systems and prym varieties in this article we consider the generalized hitchin systems introduced by bottacin and markman for an arbitrary reductive complex group g they have the structure of lagrangian fibrations pr u by generalized prym varieties over sets u parametrizing families of weyl invariant curves the g generalized hitchin systems satisfy a rank two condition and one can find invariant varieties x which distinguish these systems it is then shown that there is a correspondence between these integrable systems and the varieties x k d h which are equipped with an appropriate two form with values in the cartan subalgebra h

generalized tops and moment maps to loop algebras using a combination of hamiltonian reduction under continuous and discrete symmetry groups the lax equations for the motion of generalized tops are derived as part of a general approach to moment map embeddings of integrable hamiltonian systems in loop algebras

dualities in integrable systems and n 2 susy theories we discuss dualities in the integrable dynamics behind the exact solution to the n 2 susy yang mills theory it is shown that t duality in the string theory is related to the separation of variables procedure in the dynamical system we argue that there are analogues of s duality as well as three dimensional mirror symmetry in the many body systems of hitchin type governing the low energy effective actions

classical functional bethe ansatz for sl n separation of variables for the magnetic chain it is desirable for one to be able to construct separation variables for systems with sl n r matrices or lax pairs this article relates and extends the two known methods of construction the sklyanin s functional bethe ansatz fba commun math phys 150 181 191 1992 was previously only known for sl 2 and sl 3 and associated r matrices the algebraic geometric method of adams harnad and hurtubise commun math phys 155 385 413 1993 has been shown to work for sl n but up until now only for linear poisson brackets in this article sklyanin s program is advanced by giving the fba for certain systems with sl n r matrices this is achieved by constructing rational functions a u and b u of the matrix elements of t u so that in the generic case the zeros xi of b u are the separation coordinates and the pi a xi provide their conjugate momenta it is shown that the separation variables thus defined are the same as those given as the simultaneous solutions of several equations in two

hilbert schemes separated variables and d branes we explain sklyanin s separation of variables in geometrical terms and construct it for hitchin and mukai integrable systems we construct hilbert schemes of points on t for c c or elliptic curve and on c 2 and show that their complex deformations are integrable systems of calogero sutherland moser type we present the hyperkahler quotient constructions for hilbert schemes of points on cotangent bundles to the higher genus curves utilizing the results of hurtubise kronheimer and nakajima finally we discuss the connections to physics of d branes and string duality

force stepping integrators in lagrangian mechanics we present a class of integration schemes for lagrangian mechanics referred to as energy stepping integrators that are momentum and energy conserving symplectic and convergent in order to achieve these properties we replace the original potential energy by a piecewise constant or terraced approximation at steps of uniform height by taking steps of diminishing height an approximating sequence of energies is generated the trajectories of the resulting approximating lagrangians can be characterized explicitly and consist of intervals of piecewise rectilinear motion we show that the energy stepping trajectories are symplectic exactly conserve all the momentum maps of the original system and subject to a transversality condition converge to trajectories of the original system when the energy step is decreased to zero these properties the excellent long term behavior of energy stepping and its automatic time step selection property are born out by selected examples of application including the dynamics of a frozen argon cluster the spinning of an elastic cube and the collision of two elastic spheres

surfaces and the sklyanin bracket we discuss the lie poisson group structures associated to splittings of the loop group lgl n due to sklyanin concentrating on the finite dimensional leaves of the associated poisson structure we show that the geometry of the leaves is intimately related to a complex algebraic ruled surface with a invariant poisson structure in particular sklyanin s lie poisson structure admits a suitable abelianisation once one passes to an appropriate spectral curve the sklyanin structure is then equivalent to one considered by mukai tyurin and bottacin on a moduli space of sheaves on the poisson surface the abelianization procedure gives rise to natural darboux coordinates for these leaves as well as separation of variables for the integrable hamiltonian systems associated to invariant functions on the group

discontinuous variational time integrators for complex multibody collisions the objective of the present work is to formulate a new class of discontinuous variational time integrators that allow the system to adopt two possibly different configurations at each sampling time tk representing predictor and corrector configurations of the system the resulting sequence of configuration pairs then represents a discontinuous or non classical trajectory continuous or classical trajectories are recovered simply by enforcing a continuity constraint at all times in particular in systems subject to one sided contact constraints simulated via discontinuous variational time integrators the predictor configuration is not required to satisfy the one sided constraints whereas the corrector configuration is obtained by a closest point projection cpp onto the admissible set the resulting trajectories are generally discontinuous or non classical but are expected to converge to classical or continuous solutions for decreasing time steps we account for dissipation including friction by means of a discrete lagrange d alembert principle and make extensive use of the spacetime formalism in order to ensure exact energy conservation in conservative systems and the right rate of energy decay in dissipative systems the structure range and scope of the discontinuous variational time integrators and their accuracy characteristics are illustrated by means of examples of application concerned with rigid multibody dynamics

time discretized variational formulation of non smooth frictional contact the present work extends the non smooth contact class of algorithms introduced by kane et al to the case of friction the formulation specifically addresses contact geometries e g involving multiple collisions between tightly packed non smooth bodies for which neither normals nor gap functions can be properly defined a key aspect of the approach is that the incremental displacements follow from a minimum principle the objective function comprises terms which account for inertia strain energy contact friction and external forcing the euler lagrange equations corresponding to this extended variational principle are shown to be consistent with the equations of motion of solids in frictional contact in addition to its value as a basis for formulating numerical algorithms the variational framework offers theoretical advantages as regards the selection of trajectories in cases of non uniqueness we present numerical and analytical examples which demonstrate the good momentum and energy conservation characteristics of the numerical algorithms as well as the ability of the approach to account for stick and slip conditions

kowalewski s asymptotic method kac moody lie algebras and regularization we use an effective criterion based on the asymptotic analysis of a class of hamiltonian equations to determine whether they are linearizable on an abelian variety i e solvable by quadrature the criterion is applied to a system with hamiltonian h 1 mathord left vphantom 1 2 right kern nulldelimiterspace 2 sum limits i 1 l p i 2 sum limits i 1 l 1 exp left sum limits j 1 l n ij x j right parametrized by a real matrixn n ij of full rank it will be solvable by quadrature if and only if for alli j 2 n nt ij n n t jj 1 is a nonpositive integer i e the interactions correspond to the toda systems for the kac moody lie algebras the criterion is also applied to a system of gross neveu

isospectral hamiltonian flows in finite and infinite dimensions the approach to isospectral hamiltonian flow introduced in part i is further developed to include integration of flows with singular spectral curves the flow on finite dimensional ad invariant poisson submanifolds of the dual widetilde gl r of the positive part of the loop algebra widetilde gl r is obtained through a generalization of the standard method of linearization on the jacobi variety of the invariant spectral curves these curves are embedded in the total space of a line bundlet p1 c allowing an explicit analysis of singularities arising from the structure of the image of a moment map tilde j r m n r times m n r to widetilde gl r from the space of rank r deformations of a fixedn n matrixa it is shown how the linear flow of line bundles e t to tilde s over a suitably desingularized curve tilde s may be used to determine both the flow of matricial polynomialsl and the hamiltonian flow in the spacemn r mn r in terms of functions the resulting flows are proved to be completely integrable the reductions to subalgebras developed in part i are shown to correspond to invariance of the spectral curves and line bundles e t to tilde s under certain linear or anti linear involutions the integration of two examples from part i is given to illustrate the method the rosochatius system and the cnls coupled non linear schrodinger equation

generalized lowen functors according to their value ranges l topological spaces form different categories clearly the investigation on their relationships is certainly important and necessary lowen was one of the first authors who had studied the relation between the category of i topological spaces and that of topological spaces he introduced two well known functors and i later these functors namely lowen functors were extended by different authors for various kinds of lattices studying the relation between l top and top in this paper we introduce two functors f and if between l1 top and l2 top for each scott continuous mapping f l2 l1 where l1 and l2 are arbitrary two completely distributive lattices some topological properties related to these functors are revealed e g for li topological space xi i i 1 2 both f x1 1 and if x2 2 are lowen spaces in the case that f is the identity mapping on a linearly ordered complete lattice for l topological space x f is the finest lowen topology on x contained in and if is the coarsest lowen topology on x containing etc

smb collision detection based on temporal coherence the paper presents a novel collision detection algorithm termed the sort moving boxes smb for large number of moving 2d 3d objects which are represented by their axis aligned bounding boxes aabbs the main feature of the algorithm is the full exploitation of the temporal coherence of the objects exhibited in a dynamic environment in the algorithm the aabbs are first projected to each cartesian axis the projected intervals on the axes are separately sorted by the diminishing increment sort dis and further divided into subsections by processing all the intervals within the subsections to check if they overlap a complete contact list can be built the smb is a fast and robust collision detection algorithm particularly for systems involving a large number of moving aabbs and also supports for the dynamic insertion and deletion of objects its performance in terms of both expected total detection time and memory requirements is proportional to the total number of aabbs n and is not influenced by size differences of aabbs the space size and packing density over a large range up to ten times difference the only assumption made is that the sorted list at one time step will remain an almost sorted list at the next time step which is valid for most applications whose movement and deformation of each aabb and the dynamic change of the total number n are approximately continuous

a linear programming based algorithm for the signed separation of non smooth convex bodies a subdifferentiable global contact detection algorithm the supporting separating hyperplane ssh algorithm based on the signed distance between supporting hyperplanes of two convex sets is developed it is shown that for polyhedral sets the ssh algorithm may be evaluated as a linear program and that this linear program is always feasible and always subdifferentiable with respect to the configuration variables which define the constraint matrix this is true regardless of whether the program is primal degenerate dual degenerate or both the subgradient of the ssh linear program always lies in the normal cone of the closest admissible configuration to an inadmissible contact configuration in particular if a contact surface exists the subgradient of the ssh linear program is orthogonal to the contact surface as required of contact reactions this property of the algorithm is particularly important in modeling stiff systems rigid bodies and tightly packed or jammed systems

impact analysis of arbitrarily shaped bodies using a finite element method and a smoothed particle hydrodynamics method summary we propose a new method to obtain contact forces under a non smoothed contact problem between arbitrarily shaped bodies which are discretized by finite element method fem contact forces are calculated by the specific contact algorithm between two particles of smoothed particle hydrodynamics sph which is a meshfree method and that are applied to each colliding body this approach has advantages that accurate contact forces can be obtained within an accelerated collision without a jump problem in a discrete time increment also this can be simply applied into any contact problems like a point to point a point to line and a point to surface contact for complex shaped and deformable bodies in order to describe this method an impulse based method a unilateral contact method and sph method are firstly introduced in this paper then a procedure about the proposed method is handled detaily finally accuracy of the proposed method is verified by a conservation of momentum through three contact examples this article is protected by copyright all rights reserved

separation of variables in the classical integrable sl 3 magnetic chain there are two fundamental problems studied by the theory of hamiltonian integrable systems integration of equations of motion and construction of action angle variables a third problem however should be added to the list separation of variables though much simpler thant the two others it has important relations to quantum integrability separation of variables is constructed for thesl 3 magnetic chain an example of an integrable model associated to a nonhyperelliptic algebraic curve

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

isospectral hamiltonian flows in finite and infinite dimensions i generalized moser systems and moment maps into loop algebras a moment map jr jia gl r is constructed from the poisson manifold j a of rank r perturbations of a fixed n xn matrix a to the dual r f e positive part of the formal loop algebra i r g r c 1 the adler kostant symes theorem is used to give hamiltonians which generate commutative isospectral flows on g r the pull back of these hamiltonians by the moment map gives rise to commutative isospectral hamiltonian flows in j a the latter may be identified with flows on finite dimensional coadjoint orbits in g and linearized on the jacobi variety of an invariant spectral curve xr which generically is an r sheeted riemann surface reductions of j a are derived corresponding to subalgebras of gl r c and sl r c determined as the fixed point set of automorphism groupes generated by involutions i e jdl the classical algebras as well as reductions to twisted subalgebras of s r c the theory is illustrated by a number of examples of finite dimensional isospectral flows defining integrable hamiltonian systems and their embeddings as finite gap solutions to integrable systems of pde s

darboux coordinates and liouville arnold integration in loop algebras darboux coordinates are constructed on rational coadjoint orbits of the positive frequency part tilde mathfrak g of loop algebras these are given by the values of the spectral parameters at the divisors corresponding to eigenvector line bundles over the associated spectral curves defined within a given matrix representation a liouville generating function is obtained in completely separated form and shown through the liouvile arnold integration method to lead to the abel map linearization of all hamiltonian flows induced by the spectral invariants as illustrative examples the caseg sl 2 together with its real forms is shown to reproduce the classical integration methods for finite dimensional systems defined on quadrics with the liouville generating function expressed in hyperellipsoidal coordinates forg sl 3 the method is applied to the computation of quasi periodic solutions of the two component coupled nonlinear schrodinger equation a case which requires further symplectic constraints in order to deal with singularities in the spectral data at

finite element analysis of nonsmooth contact this work develops robust contact algorithms capable of dealing with complex contact situations involving several bodies with corners amongst the mathematical tools we bring to bear on the problem is nonsmooth analysis following clarke f h clarke optimization and nonsmooth analysis john wiley and sons new york 1983 we specifically address contact geometries for which both the use of normals and gap functions have difficulties and therefore precludes the application of most contact algorithms proposed to date such situations arise in applications such as fragmentation where angular fragments undergo complex collision sequences before they scatter we demonstrate the robustness and versatility of the nonsmooth contact algorithms developed in this paper with the aid of selected two and three dimensional applications

fuzzy topology with respect to continuous lattices abstract fuzzy topology with respect to a continuous lattice generalises the classical theory continuous functions from a topological space to a lattice with the scott topology constitute a fuzzy topology corresponding to that of lower semi continuous functions with respect to the closed unit interval fuzzy set membership is conveniently represented by the way below relation

higher separation axioms in l topologically generated i l topological spaces the main result of this paper is a theorem about inserting a pair of semicontinuous l real valued functions which extends the insertion theorem of kubiak comment math univ carolinae 34 1993 357 362 from l 0 1 to an arbitrary meet continuous lattice l endowed with an order reversing involution with this result it is shown that the normality type separation axioms in top l are preserved by the functor which takes an l topological space x to the i l topological space l x obtained by providing the set x with the i l topology consisting of all lower semicontinuous functions from x to i l the same is proved for the case of the regularity axiom

multi hamiltonian structures for r matrix systems for the rational elliptic and trigonometric r matrices we exhibit the links between three levels of poisson spaces a some finite dimensional spaces of matrix valued meromorphic functions on the complex line b spaces of spectral curves and sheaves supported on them and c symmetric products of a surface we have at each level a linear space of compatible poisson structures and the maps relating the levels are poisson this leads in a natural way to the nijenhuis coordinates for these spaces at level b there are hamiltonian systems on these spaces which are integrable for each poisson structure in the family and which are such that the lagrangian leaves are the intersections of the symplectic leaves over the poisson structures in the family specific examples include many of the well known integrable systems

the plancherel theorem for fourier laplace nahm transform for connections on the projective line we prove that the fourier laplace nahm transform for connections with finitely many logarithmic singularities and a double pole at infinity on the projective line all with semi simple singular parts is a hyper kahler isometry

galois connections between categories of l topological spaces a galois connection between two concrete categories a and b is a pair of concrete functors f email protected b g email protected a such that id y fg y y email protected b is a natural transformation from the functor email protected g to the identity functor on b and id x email protected gf x email protected a is a natural transformation from the identity functor on a email protected f in this paper it is demonstrated that for any complete lattices l 1 l 2 every galois connection between the category l 1 top of l 1 topological spaces and the category l 2 top of l 2 topologically spaces is determined by an l 1 topology d on l 2 several examples of such galois connections are given under a mild assumption it is showed that every element of the l 1 topology d is order preserving

separating coordinates for the generalized hitchin systems and the classical r matrices we exhibit natural darboux coordinates for the generalized hitchin systems studied by bottacin and markman these systems are defined on spaces of stable pairs consisting of a vector bundle and a form valued meromorphic endomorphism of the bundle in special cases genus zero genus one the bundles are rigid and one has the rational trigonometric and elliptic gaudin systems explicit formulae are given in these cases

planar dimers and harnack curves in this article we study the connection between dimers and harnack curves discovered in 15 we prove that every harnack curve arises as a spectral curve of some dimer model we also prove that the space of harnack curves of given degree is homeomorphic to a closed octant and that the areas of the amoeba holes and the distances between the amoeba tentacles give these global coordinates we characterize harnack curves of genus zero as spectral curves of isoradial dimers and also as minimizers of the volume under their ronkin function with given boundary conditions