new code upper bounds from the terwilliger algebra and semidefinite programming we give a new upper bound on the maximum size a n d of a binary code of word length n and minimum distance at least d it is based on block diagonalizing the terwilliger algebra of the hamming cube the bound strengthens the delsarte bound and can be calculated with semidefinite programming in time bounded by a polynomial in n we show that it improves a number of known upper bounds for concrete values of n and d from this we also derive a new upper bound on the maximum size a n d w of a binary code of word length n minimum distance at least d and constant weight w again strengthening the delsarte bound and yielding several improved upper bounds for concrete values of n d and w

prismatic grid generation for three dimensional complex geometries the paper describes generation of semiunstructu red prismatic grids for three dimensional complex geometries a new method of generation is presented which couples both algebraic and elliptic steps this combination yields low computation cost as well as smoothness of the grid main advantages of the developed grid generator are its direct control of grid orthogonality and spacing as well as its generality for treatment of three dimensional geometries the method marches a surface consisting of triangular faces away from the body surface employment of prismatic elements is a relatively new approach toward three dimensional complex geometry numerical simulations of viscous flows a body surface is covered with triangles which gives the geometric flexibility whereas the structure of the mesh in the direction normal to the surface provides better resolution of the viscous regions an f 16a aircraft geometry was included in the applications to investigate efficiency and to demonstrate robustness of the method in handling relatively complex topologies grid generation time for the entire aircraft domain was of the order of minutes on a workstation

numerical grid generation in arbitrary surfaces through a second order differential geometric model abstract in this paper a set of second order partial differential equations for the generation of coordinates in a given surface has been developed and then solved numerically to demonstrate its potential for surface coordinate generation the proposed equations are not some arbitrarily chosen equations but are a consequence of the formulae of gauss and thus carry with them an explicit dependence on the geometric properties of the given surface furthermore these equations are easy to solve and require only the specification of the bounding curves to provide the dirichlet boundary conditions for their solution results of coordinate generation both in simply and doubly connected regions on some known surfaces with the option of coordinate redistribution have been presented extension of this technique to arbitrary surfaces seems to be straightforward

copositive programming motivated bounds on the stability and the chromatic numbers the lovasz theta number of a graph g can be viewed as a semidefinite programming relaxation of the stability number of g it has recently been shown that a copositive strengthening of this semidefinite program in fact equals the stability number of g we introduce a related strengthening of the lovasz theta number toward the chromatic number of g which is shown to be equal to the fractional chromatic number of g solving copositive programs is np hard this motivates the study of tractable approximations of the copositive cone we investigate the parrilo hierarchy to approximate this cone and provide computational simplifications for the approximation of the chromatic number of vertex transitive graphs we provide some computational results indicating that the lovasz theta number can be strengthened significantly toward the fractional chromatic number of g on some hamming graphs

algebraic turbulence modeling for unstructured and adaptive meshes an algebraic turbulence model based on the baldwin lomax model has been implemented for use on unstructured grids the implementation is based on the use of local background turbulence meshes at each time step flow variables are interpolated from the unstructured mesh onto the background structured meshes the turbulence model is executed on these meshes and the resulting eddy viscosity values are interpolated back to the unstructured mesh modifications to the algebraic model were required to enable the treatment of more compli cated flows such as confluent boundary layers and wakes the model is used in conjunction with an efficient unstructured multigrid finite element navier stokes solver in order to compute compressible turbulent flows on fully unstructured meshes solutions about single and multiple element airfoils are obtained and compared with experimental data he use of unstructured mesh techniques for computa tional fluid dynamics cfd problems has become more widespread in recent years due to the flexibility they afford in discretizing arbitrarily complex geometries and due to the possibility they offer in resolving highly localized flow phenomena through the use of adaptive meshing however research on unstructured mesh techniques for cfd has con centrated almost exclusively on the solution of the euler equa tions in two or three dimensions for viscous flow calculations about nonsimple geometries hybrid meshes have generally been employed 1 3 where a thin structured mesh is placed in the boundary layer and wake regions and an unstructured mesh is constructed in the outer inviscid region of the flowfield besides leading to an increase in coding complexity this type of compromise limits the generality of the unstruc tured mesh approach in dealing with arbitrarily complex geometries such as multiple body geometries with close toler ances where confluent boundary layers or wakes may occur and complicates the task of performing adaptive meshing in the inviscid as well as viscous regions of flow it appears that the difficulties associated with generating highly stretched unstructured meshes which are required for efficiently resolv ing viscous shear layers as well as the efficient implementa tion of a turbulence model on such meshes has generally im peded the use of fully unstructured meshes for viscous flows the use of unstructured meshes throughout the entire flowfield is advocated in the present work previous work by the author has shown how a highly stretched unstructured mesh suitable for high reynolds number viscous flow calcula tions may be constructed4 and has also discussed the develop ment of an efficient unstructured navier stokes solver for laminar flows 5 this paper is thus concerned with the effi cient implementation of a turbulence model for computing high reynolds number turbulent flows using fully unstruc tured meshes

prediction of hysteresis associated with the static stall of an airfoil introduction t he basic phenomenon of stall associated with airfoils is quite well understood and has now become standard textbook material it is caused by massive flow separation resulting in sharp drop in lift and increase in the drag acting on the airfoil in certain cases hysteresis in the flow has been observed for angles of attack close to the stall angle however this phenomenon is not very well understood hoffmannl has reported the hysteresis loop in the data for aerodynamic coefficients for a naca 0015 airfoil he also studied the effect of freestream turbulence fst on the performance characteristics of the airfoil the hysteresis in the data can be observed for low fst but disappears for high fst the present work is an effort to study the behavior of the flow near stall by solving the governing flow equations numerically carefully conducted omputations are utilized to track the hysteresis loop in the aerodynamic data close to the stall angle to the best of the knowledge of these authors this is the first effort of its nature the incompressible reynoldsaveraged navier stokes rans equations in conjunction with the baldwin lomax moder for turbulence closure are solved using stabilized finite element formulations the finite element mesh consists of a structured mesh close to the body and an unstructured part generated via delaunay s triangulation away from the body this type of a grid has the ability of handling fairly complex geometries while still providing the desired resolution close to the body to effectively capture the boundary layer flow especially in the context of unsteady flows despite the simplicity of the baldwin lomax model its implementation with unst ctured grids is not trivial the interested reader is referred to the articles by kallinderis3 and mavriplis4 for details the finite element formulations and their implementations used in the present work are well proven and have been utilized to

solving max cut to optimality by intersecting semidefinite and polyhedral relaxations we present a method for finding exact solutions of max cut the problem of finding a cut of maximum weight in a weighted graph we use a branch and bound setting that applies a dynamic version of the bundle method as bounding procedure this approach uses lagrangian duality to obtain a nearly optimal solution of the basic semidefinite max cut relaxation strengthened by triangle inequalities the expensive part of our bounding procedure is solving the basic semidefinite relaxation of the max cut problem which has to be done several times during the bounding process we review other solution approaches and compare the numerical results with our method we also extend our experiments to instances of unconstrained quadratic 0 1 optimization and to instances of the graph equipartition problem the experiments show that our method nearly always outperforms all other approaches in particular for dense graphs where linear programming based methods fail our method performs very well exact solutions are obtained in a reasonable time for any instance of size up to n 100 independent of the density for some problems of special structure we can solve even larger problem classes we could prove optimality for several problems of the literature where to the best of our knowledge no other method is able to do so

on minimizing the spectral width of graph laplacians and associated graph realizations extremal eigenvalues and eigenvectors of the laplace matrix of a graph form the core of many bounds on graph parameters and graph optimization problems for example the value of a uniform sparsest cut is bounded from below and above by the second smallest and the largest eigenvalue of the weighted laplacian divided by the number of nodes of the graph minimizing the difference between maximum and second smallest eigenvalue over edge weighted laplacians of a graph reduces the size of this interval we study this problem of minimizing the spectral width for its own sake with the goal of advancing the understanding of connections between structural properties of the graph and the corresponding eigenvectors and eigenvalues building on previous work where these eigenvalues were investigated separately we show that a corresponding dual problem allows us to view eigenvectors to optimized eigenvalues as graph realizations in euclidean space whose structure is tightly linked to the separator structure of the g

on the decoupling of the homogeneous and inhomogeneous parts in inhomogeneous quantum groups we show that if there exists a realization of a hopf algebra h in a h module algebra then one can split their cross product into the tensor product algebra of itself with a subalgebra isomorphic to h and commuting with this result applies in particular to the algebra underlying inhomogeneous quantum groups like the euclidean groups which are obtained as cross products of the quantum euclidean spaces qn with the quantum groups of rotation uq so n of qn for which it has no classical analogue

hybrid prismatic tetrahedral grid generation for viscous flows around complex geometries a method for the generation of hybrid prismatic tetrahedral grids is described for complex three dimensional geometries including multibody domains and employed for viscous flows around an aircraft the prisms cover the region close to each body s surface while tetrahedra are created elsewhere a special method is presented that allows the generation of single block nonoverlapping prismatic meshes even in complex geometries that contain narrow gaps and cavities examples of cases treated by this method include wing engine configurations as well as multi element wings the second development is a combined octree advancing front method for the generation of the tetrahedra of the hybrid mesh the main feature of this octree based tetrahedra generator is that it does not require the creation of a background mesh by the user for the determination of the grid spacing and stretching parameters these are determined via an automatically generated octree the hybrid grid generator is robust geometry independent and requires no user intervention the suitability of hybrid meshes for capturing viscous flow phenomena is demonstrated by simulation of viscous flows around a high speed civil transport type of aircraft configuration reduction in computer resources has been substantial allowing flow simulations to be performed on workstations rather than supercomputers

reduction of symmetric semidefinite programs using the regular ast representation we consider semidefinite programming problems on which a permutation group is acting we describe a general technique to reduce the size of such problems exploiting the symmetry the technique is based on a low order matrix representation of the commutant centralizer ring of the matrix algebra generated by the permutation matrices we apply it to extending a method of de klerk et al that gives a semidefinite programming lower bound to the crossing number of complete bipartite graphs it implies that cr k 8 n 2 9299n 2 6n cr k 9 n 3 8676n 2 8n and for any m 9 lim n to infty frac rm cr k m n z m n geq 0 8594 frac m m 1 where z m n is the zarankiewicz number lfloor frac 1 4 m 1 2 rfloor lfloor frac 1 4 n 1 2 rfloor which is the conjectured value of cr k m n here the best factor previously known was 0 8303 instead of 0 8594

unbounded elements affiliated with c algebras and non compact quantum groups the affiliation relation that allows to include unbounded elements operators into thec algebra framework is introduced investigated and applied to the quantum group theory the quantum deformation of the two fold covering of the group of motions of euclidean plane is constructed a remarkable radius quantization is discovered it is also shown that the quantumsu 1 1 group does not exist on thec algebra level for real value of the deformation parameter

numerical block diagonalization of matrix algebras with application to semidefinite programming semidefinite programming sdp is one of the most active areas in mathematical programming due to varied applications and the availability of interior point algorithms in this paper we propose a new pre processing technique for sdp instances that exhibit algebraic symmetry we present computational results to show that the solution times of certain sdp instances may be greatly reduced via the new approach

finite element method for incompressible viscous flows with adaptive hybrid grids a new adaptive finite element numerical method has been developed for the unsteady navier stokes equations of incompressible flow in two dimensions the momentum equations combined with a pressure correction equation are solved employing a nonstaggered grid the solution is advanced in time with an explicit implicit marching scheme an adaptive algorithm has been implemented which refines the grid locally to resolve detected flow features a combination of quadrilateral as well as triangular cells provides a stable and accurate numerical treatment of grid interfaces that are located within regions of high gradients applications of the developed adaptive algorithm include both steady and unsteady flows with low and high reynolds numbers comparisons with analytical as well as experimental data evaluate accuracy and robustness of the method i introduction i ncompressible flows are frequently encountered in engineering applications during the past two decades a significant number of numerical algorithms have been developed for solution of the incompressible navier stokes equations 1 liie lack of pressure term in the continuity equation makes solution of the momentum equations with the divergence free constraint more difficult in the case of incompressible flows the conservation of mass acts as a constraint condition that the velocity field must satisfy whereas in compressible flows the conservation of mass is given through a partial differential equation for the temporal variation of density the infinite speed of sound in the incompressible case requires an implicit treatment of the pressure furthermore spatial discretization for pressure and velocity may produce oscillatory solutions one approach followed is to formulate the equations in terms of a stream function and a vorticity extension of this method to three dimensions is not possible different formulations have been used in three dimensions such as the vorticity velocity approach another approach is to use the compressible flow equations and solve them for low mach numbers the required time step for such computations is very small because the speed of sound approaches infinity at the incompressible limit a method that uses compressible like governing equations is the artificial compressibility approach 2 a time derivative of the pressure is added to the continuity equation and the incompressible flowfield is treated as compressible during the transient stage time accuracy of the simulation is usually not preserved another class of algorithms uses a special poisson equation for the pressure field 5 8 the usual computational procedure is to assume an initial pressure field and then an iterative process is defined until the continuity equation is satisfied a major issue of the corresponding pressure and velocity spatial discretization is oscillations in the pressure field to reject these modes staggered grids have been employed by several of these algorithms 9 10 on the other hand employment of nonstaggered grids11 13 requires dissipation in the algorithms stability of both approaches with high reynolds number flows is an important issue a review of numerical methods for incompressible flows as well as references

strengthened semidefinite programming bounds for codes we give a hierarchy of semidefinite upper bounds for the maximum size a n d of a binary code of word length n and minimum distance at least d at any fixed stage in the hierarchy the bound can be computed to an arbitrary precision in time polynomial in n this is based on a result of de klerk et al math program 2006 about the regular representation for matrix algebras the delsarte bound for a n d is the first bound in the hierarchy and the new bound of schrijver ieee trans inform theory 51 2859 2866 2005 is located between the first and second bounds in the hierarchy while computing the second bound involves a semidefinite program with o n7 variables and thus seems out of reach for interesting values of n schrijver s bound can be computed via a semidefinite program of size o n3 a result which uses the explicit block diagonalization of the terwilliger algebra we propose two strengthenings of schrijver s bound with the same computational complexity

automatic adaptive grid refinement for the euler equations a method of adaptive grid refinement for the solution of the steady euler equations for transonic flow is presented the algorithm automatically decides where the coarse grid accuracy is insufficient and creates locally uniform refined grids in these regions this typically occurs at the leading and trailing edges the solution is then integrated to steady state using the same integrator flo52 in the interior of each grid the boundary conditions needed on the fine grids are examined and the importance of treating the fine coarse grid interface conservatively is discussed numerical results indicate substantial computational savings for the same solution accuracy can be achieved

exploiting special structure in semidefinite programming a survey of theory and applications abstract semidefinite programming sdp may be seen as a generalization of linear programming lp in particular one may extend interior point algorithms for lp to sdp but it has proven much more difficult to exploit structure in the sdp data during computation we survey three types of special structures in sdp data 1 a common chordal sparsity pattern of all the data matrices this structure arises in applications in graph theory and may also be used to deal with more general sparsity patterns in a heuristic way 2 low rank of all the data matrices this structure is common in sdp relaxations of combinatorial optimization problems and sdp approximations of polynomial optimization problems 3 the situation where the data matrices are invariant under the action of a permutation group or more generally where the data matrices belong to a low dimensional matrix algebra such problems arise in truss topology optimization particle physics coding theory computational geometry and graph theory we will give an overview of existing techniques to exploit these structures in the data most of the paper will be devoted to the third situation since it has received the least attention in the literature so far

a two parameter quantum deformation of the unit disc we define and study a two parameter deformation of the unit disc this deformation is described in terms of a family of type i c algebras c q we study the representation theory for c q and construct an action of the quantized universal enveloping algebra uq sl 2 on c q

solving semidefinite quadratic linear programs using sdpt3 this paper discusses computational experiments with linear optimization problems involving semidefinite quadratic and linear cone constraints sqlps many test problems of this type are solved using a new release of sdpt3 a matlab implementation of infeasible primal dual path following algorithms the software developed by the authors uses mehrotra type predictor corrector variants of interior point methods and two types of search directions the hkm and nt directions a discussion of implementation details is provided and computational results on problems from the sdplib and dimacs challenge collections are reported

quantum field theory on the noncommutative plane with eq 2 symmetry we study properties of a scalar quantum field theory on the two dimensional noncommutative plane with eq 2 quantum symmetry we start from the consideration of a firstly quantized quantum particle on the noncommutative plane then we define quantum fields depending on noncommutative coordinates and construct a field theoretical action using the eq 2 invariant measure on the noncommutative plane with the help of the partial wave decomposition we show that this quantum field theory can be considered as a second quantization of the particle theory on the noncommutative plane and that this field theory has contrary to the common belief even more severe ultraviolet divergences than its counterpart on the usual commutative plane finally we introduce the symmetry transformations of physical states on noncommutative spaces and discuss them in detail for the case of the eq 2 quantum group

algebraic turbulence modeling for adaptive unstructured grids algebraic turbulence models are quite common in flow numerical simulations their simplicity and low computation cost as well as their applicability to specific classes of flows have made them attractive compared to the more expensive multiple equation models adaptive unstructured grid algorithms have become quite popular over the last years due to their flexibility in treating complex geometries as well their high resolution capability for local flow phenomena however application of algebraic turbulence models with unstructured grids proves to be nontrivial the difficulties associated with implementation of those models with such grids have generally impeded use of unstructured grids for turbulent flow simulations this paper presents a novel method of implementing an algebraic turbulence model with embedded unstructured grids the baldwinlomax model was chosen in order to test the implementation modifications to the application of the original model on structured grids are suggested as well comparisons with both experiments and other numerical simulations are employed in order to evaluate accuracy of the implementation method onsiderable progress has been made in the development of numerical methods for the solution of the navier stokes equations however most of those methods are impractical for complicated flows in a design environment the primary reason is poor efficiency which makes it difficult to obtain accurate results very fine resolution is needed which results in long computation times even with the use of available supercomputers in general the selection of the equations to be solved the scheme and the grid all are determined a priori by the user before starting the solution procedure and quite often some and even all of these factors must be modified by the user in order to improve the results the robustness of current numerical schemes as well as present computer capabilities has recently allowed a dramatic change in this philosophy general algorithms have been developed that are flexible enough to adaptively adjust those parameters during the solution procedure without intervention by the user one of the most efficient adaptive methods consists of local refinement by dividing grid cells an initially coarse grid is embedded in regions with large flow gradients e g boundary layers shocks wakes etc the algorithm senses high gradient regions and automatically divides grid cells in such regions this approach has been used for the resolution of shocks in flowfields described by euler equations 1 2 the embedded scheme was used as part of an adaptive algorithm for turbulent flows 3 6 algebraic turbulence models are quite common in aerodynamics numerical simulations their simplicity and low computation cost as well as their accuracy for specific classes of flows have made them attractive compared to the more expensive multiple equation models however such models require nonlocal information that characterizes an entire shearlayer profile the adaptive algorithm results in an unstructured mesh which is a major issue in the implementation of an algebraic turbulence model grid interfaces frequently interrupt normal to wall meshlines and therefore models that employ information from an entire shear layer profile cannot be applied similarly in the case of triangular meshes such normal grid lines generally do not exist a new method of

generalized vortex lattice method for planar supersonic flow the generalized vortex lattice method for oscillating lifting surfaces in subsonic flow is extended to the supersonic planar regime

unitary representations of the q oscillator algebra we classify the unitary irreducible representations of the q oscillator algebra bb q2b b 1 by the sign of the definite operator k bb b b k 0 corresponds to fock representations and k 0 to non fock ones k 0 being a degenerate case we present their link to representations of extended q oscillator algebra aa qa a q n obtained earlier we give examples of all representations besides standard ones we describe a new non fock representation of the macfarlane type

the euclidean hopf algebra u q e n and its fundamental hilbert space representations the euclidean hopf algebra uq en dual of fun rnq soq 1 n is constructed by realizing it as a subalgebra of the differential algebra diff rnq on the quantum euclidean space rnq in fact the previous realization g fiore commun math phys 169 475 500 1995 of uq 1 so n is extended within diff rnq through the introduction of q derivatives as generators of q translations the fundamental hilbert space representations of uq en turn out to be of highest weight type and rather simple lattice regularized versions of the classical ones the vectors of a basis of the singlet i e zero spin irrep can be realized as normalizable functions on rnq going to distributions in the limit q 1

hysteresis in flow past a naca 0012 airfoil computations for two dimensional flow past a stationary naca 0012 airfoil are carried out with progressively increasing and decreasing angles of attack the incompressible reynolds averaged navier stokes equations in conjunction with the baldwin lomax model for turbulence closure are solved using stabilized finite element formulations beyond a certain angle of attack the flow stalls with a sudden loss of lift and increase in drag hysteresis in the aerodynamic coefficients is observed for a small range of angles of attack close to the stall angle this is caused by the difference in the location of the separation point of the flow on the upper surface of the airfoil during the increasing and decreasing angles of attack with the increasing angle the separation point moves gradually towards the leading edge with the decreasing angle the movement of the separation point away from the leading edge is abrupt

copositive optimization recent developments and applications due to its versatility copositive optimization receives increasing interest in the operational research community and is a rapidly expanding and fertile field of research it is a special case of conic optimization which consists of minimizing a linear function over a cone subject to linear constraints the diversity of copositive formulations in different domains of optimization is impressive since problem classes both in the continuous and discrete world as well as both deterministic and stochastic models are covered copositivity appears in local and global optimality conditions for quadratic optimization but can also yield tighter bounds for np hard combinatorial optimization problems here some of the recent success stories are told along with principles algorithms and applications

hilbert space representations of cross product algebras in this paper we study and classify hilbert space representations of cross product ast algebras of the quantized enveloping algebra user1 mathcal u q left text e text 2 right with the coordinate algebras user1 mathcal o left text e q left 2 right right of the quantum motion group and user1 mathcal o left mathbb c q right of the complex plane and of the quantized enveloping algebra user1 mathcal u q left text su 1 1 right with the coordinate algebras user1 mathcal o left text su q left 1 1 right right of the quantum group mathrm su q 1 1 and user1 mathcal o left text u p right of the quantum disc invariant positive functionals and the corresponding heisenberg representations are explicitly described

adaptive refinement coarsening scheme for three dimensional unstructured meshes the paper describes the development and application of an adaptive algorithm for tetrahedral grids an initial unstructured grid is adapted by employing local division as well as deletion of the tetrahedral cells the process is dynamic and the adapted grid changes follow evolution of the solution adaptation of the cells consists of normal division of a tetrahedron into eight subcells as well as directional division into two or four subcells a major issue of such adaptive grid algorithms is elimination of hanging nodes that appear on the edges in the interface between the embedded and unembedded zones a novel technique has been developed for treatment of such interface cells that can be applied to eliminate any possible hanging node configuration in a simple way a special data structure based on octrees has been developed which is flexible to handle all different types of cell division deletion application cases include a moving source as well as transonic flow around the onera m6 wing

symmetry in semidefinite programs this paper is a tutorial in a general and explicit procedure to simplify semidefinite programs which are invariant under the action of a symmetry group the procedure is based on basic notions of representation theory of finite groups as an example we derive the block diagonalization of the terwilliger algebra of the binary hamming scheme in this framework here its connection to the orthogonal hahn and krawtchouk polynomials becomes visible

improved lower bounds for the 2 page crossing numbers of k_ m n and k_n via semidefinite programming it has long been conjectured that the crossing numbers of the complete bipartite graph k m n and of the complete graph k n equal z m n bigl lfloor frac n 2 bigr rfloor bigl lfloor frac n 1 2 bigr rfloor bigl lfloor frac m 2 bigr rfloor bigl lfloor frac m 1 2 bigr rfloor and z n frac 1 4 bigl lfloor frac n 2 bigr rfloor bigl lfloor frac n 1 2 bigr rfloor bigl lfloor frac n 2 2 bigr rfloor bigl lfloor frac n 3 2 bigr rfloor respectively in a 2 page drawing of a graph the vertices are drawn on a straight line the spine and each edge is contained in one of the half planes of the spine the 2 page crossing number nu 2 g of a graph g is the minimum number of crossings in a 2 page drawing of g somewhat surprisingly there are 2 page drawings of k m n respectively k n with exactly z m n respectively z n crossings thus yielding the conjectures i nu 2 k m n stackrel z m n and ii nu 2 k n stackrel z n it