on groups of finite morley rank with a split bn pair of rank 1 abstract we study groups of finite morley rank with a split bn pair of tits rank 1 in the case where the normal complement to b n in b is infinite and abelian for such groups we give conditions ensuring that the standard action of g on the cosets of b is isomorphic to the natural action of psl 2 f on p 1 f for f an algebraically closed field in particular we show that sl 2 f and psl 2 f are the only infinite quasisimple l groups of finite morley rank possessing a split bn pair of tits rank 1 where the normal complement to b n in b is infinite abelian our approach is through the theory of moufang sets and is tied to work attempting to classify the abelian moufang sets of finite morley rank

circular planar graphs and resistor networks abstract we consider circular planar graphs and circular planar resistor networks associated with each circular planar graph there is a set p q of pairs of sequences of boundary nodes which are connected through a graph is called critical if removing any edge breaks at least one of the connections p q in we prove that two critical circular planar graphs are y equivalent if and only if they have the same connections if a conductivity is assigned to each edge in there is a linear from boundary voltages to boundary currents called the network response this linear map is represented by a matrix we show that if is any circular planar resistor network whose underlying graph is critical then the values of all the conductors in may be calculated from finally we give an algebraic description of the set of network response matrices that can occur for circular planar resistor networks

a front remeshing technique for a lagrangian description of moving interfaces in two fluid flows the numerical analysis of two fluid flows involves the treatment of a discontinuity that appears at the separating interface classical lagrangian schemes applied to update the front position between two immiscible incompressible fluids have been long recognized to provide a sharp representation of the interface however the main drawback of these approaches is the progressive distortion in the distribution of the markers used to identify the material front to avoid this problem an interface remeshing algorithm based on the diffuse approximation of the interface curvature is proposed in this work in addition the remeshed front is enforced to preserve the global volume these new aspects are incorporated in an existing fluid dynamics formulation for the analysis of two fluid flows problems the resulting formulation is called in this work as the moving lagrangian interface remeshing technique mlirt copyright 2005 john wiley sons ltd

a face oriented stabilized nitsche type extended variational multiscale method for incompressible two phase flow a novel extended variational multiscale method for incompressible two phase flow is proposed in this approach the level set method which allows for accurately representing complex interface evolutions is combined with an extended finite element method for the fluid field sharp representation of the discontinuities at the interface related to surface tension effects and large material parameter ratios are enabled by this approach to capture the discontinuities jump enrichments are applied for both velocity and pressure field nitsche s method is then used to weakly impose the continuity of the velocity field for a stable formulation on the entire domain residual based variational multiscale terms are supported by appropriate face oriented ghost penalty and fluid stabilization terms in the region of enriched elements both face oriented terms as well as interfacial terms related to nitsche s method are introduced such that it is accounted for viscous and convection dominated transient flows as a result stability and wellconditioned systems are guaranteed independent of the interface location the proposed method is applied to four numerical examples of increasing complexity two dimensional rayleigh taylor instabilities a twodimensional collapsing water column three dimensional rising bubbles as well as a three dimensional bubble coalescence excellent agreement with either analytical solutions or numerical and experimental reference data as well as robustness for all flow regimes is demonstrated for all examples

a grid alignment finite element technique for incompressible multicomponent flows we present a numerical method for solving the multicomponent incompressible navier stokes equations the methods employs a moving grid technique and a projection scheme based on low order crouzeix raviart and p 1 finite elements the computational grid is aligned with the fluid interface at each time step to keep the interpolation order optimal the connectivity of the grid is fixed during the entire time simulation to ensure efficient assembling and parallelization the method is tested on several examples

implicit tracking for multi fluid simulations in this work we introduce a new method for tracking the interfaces among several immiscible fluids to advance the solution in time we use a muscl type scheme which couples a partial volume representation with a level set one to build the numerical fluxes in particular we show the positiveness and conservation properties of this method some numerical tests are given to demonstrate the conservativeness and the performances of our method

on the number of negative eigenvalues of the laplacian on a metric graph the number of negative eigenvalues of self adjoint laplacians on metric graphs is calculated in terms of the boundary conditions and the underlying geometric structure this extends and complements earlier results by kostrykin and schrader 2006 contemp math 415 201 25

proper moufang sets with abelian root groups are special moufang sets are split bn pairs of rank one or the moufang buildings of rank one as such they have been studied extensively being the basic building blocks of all split bn pairs a moufang set is proper if it is not sharply 2 transitive we prove that a proper moufang set whose root groups are abelian is special this resolves an important conjecture in the area of moufang sets it enables us to apply the theory of quadratic jordan division algebras to such moufang sets

rapid and accurate computation of the distance function using grids we present two fast and simple algorithms for approximating the distance function for given isolated points on uniform grids the algorithms are then generalized to compute the distance to piecewise linear objects by incorporating the geometry of huygens principle in the reverse order with the classical viscosity solution theory for the eikonal equation u 1 the algorithms become almost purely algebraic and yield very accurate approximations the generalized closest point formulation used in the second algorithm provides a framework for further extension to compute the distance accurately to smooth geometric objects on different grid geometries without the construction of the voronoi diagrams this method provides a fast and simple translator of data commonly given in computational geometry to the volumetric representation used in level set methods

numerical simulation of premixed combustion using an enriched finite element method in this paper we present a novel discretization technique for the simulation of premixed combustion based on a locally enriched finite element method fem use is made of the g function approach to premixed combustion in which the domain is divided into two parts one part containing the burned and another containing the unburned gases a level set or g function is used to define the flame interface separating burned from unburned gases the extended finite element method x fem is employed which allows for velocity and pressure fields that are discontinuous across the flame interface lagrange multipliers are used to enforce the correct essential interface conditions in the form of jump conditions across the embedded flame interface a persisting problem with the use of lagrange multipliers in x fem has been the discretization of the lagrange multipliers in this paper the distributed lagrange multiplier technique is adopted we will provide results from a spatial convergence analysis showing good convergence however a small modification of the interface is required to ensure a unique solution finally results are presented from the application of the method to the problems of moving flame fronts the darrieus landau instability and a piloted bunsen burner flame

a semi lagrangean time integration approach for extended finite element methods many computational problems incorporate discontinuities that evolve in time the extendend finite element method xfem is able to represent discontinuities sharply on fixed arbitrary meshes but numerical difficulties arise if these discontinuities move in time we point out that this issue is crucial for interface problems with strongly discontinuous fields on fixed grids a method using semi lagrangean techniques is proposed to adequately handle time integration based on finite difference schemes in the context of the xfem the basic idea is to adapt previous numerical solutions to the current interface position by tracking back virtual lagrangean particles to their previous positions where an appropriate solution can be extrapo lated from a smooth field convergence properties of the proposed method in time and space are thoroughly studied for two one dimensional model problems finally the method is applied to the particularly challenging problem of premixed combustion where the discontinuity appears at the flame front separating the burnt from the unburnt gases a two dimensional and a three dimensional expanding flame demonstrates that the method is sufficiently accurate to retain the properties of the overall nitsche type formulation for interface problems with embedded strong discontinuities

identities in moufang sets moufang sets were introduced by jacques tits as an axiomatization of the buildings of rank one that arise from simple algebraic groups of relative rank one these fascinating objects have a simple definition and yet their structure is rich while it is rigid enough to allow for at least partial classification in this paper we obtain two identities that hold in any moufang set these identities are closely related to the axioms that define a quadratic jordan algebra we apply them in the case when the moufang set is so called special and has abelian root groups in addition we push forward the theory of special moufang sets

an extended residual based variational multiscale method for two phase flow including surface tension abstract in this study an extended residual based variational multiscale method is proposed for two phase flow including surface tension the extended residual based variational multiscale method combines a residual based form of the variational multiscale method and the extended finite element method xfem by extending the solution spaces it is possible to reproduce discontinuities of the solution fields inside elements intersected by the interface in particular we propose a quasi static enrichment to reproduce time dependent discontinuities kink enrichments of both velocity and pressure as well as kink enrichments of velocity combined with jump enrichments of pressure are considered here to capture the interface between the phases on a fixed grid a level set approach is used a residual based variational multiscale method is employed for computing both flow and interface motion the presented method is tested for various two phase flow examples exhibiting small and large density and viscosity ratios with and without surface tension a two phase couette flow a rayleigh taylor instability a sloshing tank and a three dimensional rising bubble to the best of our knowledge these are the first simulation results for representative time dependent three dimensional two phase flow problems using an extended finite element method stable and accurate results are obtained for all test examples

numerical assessment of stability of interface discontinuous finite element pressure spaces abstract the stability of two recently developed pressure spaces has been assessed numerically the space proposed by ausas et al r f ausas f s sousa g c buscaglia an improved finite element space for discontinuous pressures comput methods appl mech engrg 199 2010 1019 1031 which is capable of representing discontinuous pressures and the space proposed by coppola owen and codina a h coppola owen r codina improving eulerian two phase flow finite element approximation with discontinuous gradient pressure shape functions int j numer methods fluids 49 2005 1287 1304 which can represent discontinuities in pressure gradients we assess the stability of these spaces by numerically computing the inf sup constants of several meshes the inf sup constant results as the solution of a generalized eigenvalue problems both spaces are in this way confirmed to be stable in their original form an application of the same numerical assessment tool to the stabilized equal order p 1 p 1 formulation is then reported an interesting finding is that the stabilization coefficient can be safely set to zero in an arbitrary band of elements without compromising the formulation s stability an analogous result is also reported for the mini element p 1 p 1 when the velocity bubbles are removed in an arbitrary band of elements

on multiplicity of a quantum graph spectrum the sturm liouville equations on the edges of a metric connected graph together with the boundary and matching conditions at the vertices generate a spectral problem for a self adjoint operator it is shown that if the graph is not cyclically connected then the maximal multiplicity of an eigenvalue of such an operator is gt ptin where is the cyclomatic number of the graph and gt and ptin are the number of edges and the number of interior vertices respectively for the tree obtained by contracting all the cycles of the graph into vertices if the graph is cyclically connected then the maximal multiplicity of an eigenvalue is 1

can one hear the shape of a graph we show that the spectrum of the schroperator on a finite metric graph determines uniquely the connectivity matrix and the bond lengths provided that the lengths are non commensurate and the connectivity is simple no parallel bonds between vertices and no loops connecting a vertex to itself that is one can hear the shape of the graph we also consider a related inversion problem a compact graph can be converted into a scattering system by attaching to its vertices leads to infinity we show that the scattering phase determines uniquely the compact part of the graph under similar conditions as above

no quantum ergodicity for star graphs we investigate statistical properties of the eigenfunctions of the schrodinger operator on families of star graphs with incommensurate bond lengths we show that these eigenfunctions are not quantum ergodic in the limit as the number of bonds tends to infinity by finding an observable for which the quantum matrix elements do not converge to the classical average we further show that for a given fixed graph there are subsequences of eigenfunctions which localise on pairs of bonds we describe how to construct such subsequences explicitly these structures are analogous to scars on short unstable periodic orbits

scattering theory on graphs we consider the scattering theory for the schrodinger operator dx2 v x on the graphs made of one dimensional wires connected to external leads we derive two expressions for the scattering matrix on arbitrary graphs one involves matrices that couple arcs oriented bonds the other involves matrices that couple vertices we discuss a simple way to tune the coupling between the graph and the leads the efficiency of the formalism is demonstrated with a few known examples

a fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies application to particulate flow abstract in this article we discuss a methodology that allows the direct numerical simulation of incompressible viscous fluid flow past moving rigid bodies the simulation methods rest essentially on the combination of a lagrange multiplier based fictitious domain methods which allow the fluid flow computations to be done in a fixed flow region b finite element approximations of the navier stokes equations occurring in the global model c time discretizations by operator splitting schemes in order to treat optimally the various operators present in the model the above methodology is particularly well suited to the direct numerical simulation of particulate flow such as the flow of mixtures of rigid solid particles and incompressible viscous fluids possibly non newtonian we conclude this article with the presentation of the results of various numerical experiments including the simulation of store separation for rigid airfoils and of sedimentation and fluidization phenomena in two and three dimensions

recovering a quantum graph spectrum from vertex data we study the question to what extent spectral information on a schrodinger operator on a finite compact metric graph subject to standard or type matching conditions can be recovered from a corresponding titchmarsh weyl function on the boundary of the graph in contrast to the case of ordinary or partial differential operators the knowledge of the titchmarsh weyl function is in general not sufficient for recovering the complete spectrum of the operator or the potentials on the edges however it is shown that those eigenvalues with sufficiently high depending on the cyclomatic number of the graph multiplicities can be recovered moreover we prove that under certain additional conditions the titchmarsh weyl function even contains the full spectral information

the extended finite element method for two phase and free surface flows a systematic study in immiscible two phase flows jumps or kinks are present in the velocity and pressure fields across the interfaces of the two fluids the extended finite element method xfem is able to reproduce such discontinuities within elements robust and accurate interface capturing schemes with no restrictions on the interface topology are thereby enabled this paper investigates different enrichment schemes and time integration schemes within the xfem test cases with and without surface tension on moving or stationary meshes are studied and compared to interface tracking results when possible a particularly useful setting is extracted which is recommended for two phase flows an extension of this formulation for the simulation of free surface flows and of floating objects is proposed

special moufang sets their root groups and their μ maps we prove timmesfeld s conjecture that special abstract rank one groups are quasisimple we give two characterizations of the root groups in special moufang sets a normal subgroup of the point stabilizer is a root group if it is either regular or nilpotent and transitive we prove that if a root group of a special moufang set contains an involution then it is of exponent 2 we also show that the root groups are abelian if and only if the so called maps are involutions

an improved finite element space for discontinuous pressures abstract we consider incompressible stokes flow with an internal interface at which the pressure is discontinuous as happens for example in problems involving surface tension we assume that the mesh does not follow the interface which makes classical interpolation spaces to yield suboptimal convergence rates typically the interpolation error in the l 2 norm is of order h 1 2 we propose a modification of the p 1 conforming space that accommodates discontinuities at the interface without introducing additional degrees of freedom or modifying the sparsity pattern of the linear system the unknowns are the pressure values at the vertices of the mesh and the basis functions are computed locally at each element so that the implementation of the proposed space into existing codes is straightforward with this modification numerical tests show that the interpolation order improves to o h 3 2 the new pressure space is implemented for the stable p 1 p 1 mini element discretization and for the stabilized equal order p 1 p 1 discretization assessment is carried out for poiseuille flow with a forcing surface and for a static bubble in all cases the proposed pressure space leads to improved convergence orders and to more accurate results than the standard p 1 space in addition two navier stokes simulations with moving interfaces rayleigh taylor instability and merging bubbles are reported to show that the proposed space is robust enough to carry out realistic simulations

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

a borg levinson theorem for trees we prove that the dirichelet to neumann map for a schrodinger operator on a finite simply connected tree determines uniquely the potential on that tree

a level set formulation of eulerian interface capturing methods for incompressible fluid flows a level set formulation is derived for incompressible immiscible navier stokes equations separated by a free surface the interface is identified as the zero level set of a smooth function eulerian finite difference methods based on this level set formulation are proposed these methods are robust and efficient and are capable of computing interface singularities such as merging and reconnection numerical experiments are presented to demonstrate the effectiveness of the methods

a characteristic finite element algorithm for the 3 d navier stokes equations using unstructured grids abstract this paper discusses a finite element algorithm for solution of the 3 d navier stokes equations that is based on the method of characteristics the algorithm which is based on a time slitting of the convective terms from the generalized stokes problem comprises a point wise characteristic integration of the convective terms and uzawa integration of the generalized stokes problem the most costly part of the characteristic integration scheme is the search for an element containing the foot of a given characteristic the present study uses a very efficient algorithm for element searching which is particularly useful for complex geometries the implementation of the algorithm is discussed in the paper and the resulting code s performance is compared to that of a code based on a more conventional fourth order runge kutta convection integration a variety of flow problems are tested including an analytical solution of the navier stokes equations flow in a lid driven cavity flow over a backward facing step and flow in an end to side anastomoses in all cases the proposed algorithm demonstrated performance superior to that of a conventional explicit treatment of the convective terms

moufang sets and jordan division algebras we give a simple criterion which determines when a permutation group u and one additional permutation give rise to a moufang set we apply this criterion to show that every jordan division algebra gives rise in a very natural way to a moufang set to provide sufficient conditions for a moufang set to arise from a jordan division algebra and to give a characterization of the projective moufang sets over a commutative field of characteristic different from 2

kirchhoff s rule for quantum wires we formulate and discuss one particle quantum scattering theory on an arbitrary finite graph with n open ends and where we define the hamiltonian to be minus the laplace operator with general boundary conditions at the vertices this results in a scattering theory with n channels the corresponding on shell s matrix formed by the reflection and transmission amplitudes for incoming plane waves of energy e 0 is given explicitly in terms of the boundary conditions and the lengths of the internal lines it is shown to be unitary which may be viewed as the quantum version of kirchhoff s law we exhibit covariance and symmetry properties it is symmetric if the boundary conditions are real also there is a duality transformation on the set of boundary conditions and the lengths of the internal lines such that the low energy behaviour of one theory gives the high energy behaviour of the transformed theory finally we provide a composition rule by which the on shell s matrix of a graph is factorizable in terms of the s matrices of its subgraphs all proofs use only known facts from the theory of self adjoint extensions standard linear algebra complex function theory and elementary arguments from the theory of hermitian symplectic forms

moufang sets of finite morley rank of odd type abstract we show that for a wide class of groups of finite morley rank the presence of a split bn pair of tits rank 1 forces the group to be of the form psl 2 and the bn pair to be standard our approach is via the theory of moufang sets specifically we investigate infinite and so called hereditarily proper moufang sets of finite morley rank in the case where the little projective group has no infinite elementary abelian 2 subgroups and show that all such moufang sets are standard and thus associated to psl 2 f for f an algebraically closed field of characteristic not 2 provided the hua subgroups are nilpotent further we prove that the same conclusion can be reached whenever the hua subgroups are l groups and the root groups are not simple

a fictitious domain finite element method for particulate flows the paper presents a finite element method for the direct numerical simulation of 3d incompressible fluid flows with suspended rigid particles it uses the fictitious domain approach extending the fluid domain into the domain occupied by the particles although the idea for the present formulation came from the fictitious domain formulation of glowinski et al j comput phys 169 2001 363 it finally yielded a new scheme which allows for a more efficient integration of the system of equations and which is easier to implement algorithmically it also avoids the need to grid the rigid particles and solves the entire problem on a single eulerian grid the spatial discretization is performed with p2 p1 finite elements but any other stable pair of approximations can be used the present scheme is compared to a second order scheme based on the approach in glowinski et al loc cit using the test case of a single sphere in a viscous liquid its accuracy costs performance is excellent

a finite element technique for multifluid incompressible flow using eulerian grids the paper presents a finite element method for 3d incompressible fluid flows with capillary free boundaries it uses a fixed eulerian grid of 10 nodes p2 p1 tetrahedra and tracks the free boundary using a six nodes p2 triangular surface grid in order to improve the mass conservation properties of the method a local enrichment of the finite element basis in the elements intersected by the free boundaries is employed in addition to the surface tracking it also advects a smooth indicator function for an easy identification of the fluid properties in the different parts of the domain the advective part of the navier stokes equations is split and integrated with a characteristic method the remaining generalized stokes problem is resolved by means of an inexact outer inner uzawa iteration with a properly chosen preconditioner the performance of this technique is evaluated on several problems involving droplets in viscous liquids

the m matrix inverse problem for the sturm liouville equation on graphs abstractwe consider an inverse spectral problemfor sturm liouville boundaryvalue problemson a graph with formally self adjoint boundary conditions at the nodes where the giveninformation is the m matrix based on the results found in s currie b a watson m matrix asymptotics for sturm liouville problems on graphs j com appl math doi 10 1016 j cam 2007 11 019 using the green s function we prove that the poles ofthe m matrix are at the eigenvalues of the associated boundary value problem and aresimple located on the real axis and that the residue at a pole is a negative semi de nitematrix with rank equal to the multiplicity of the eigenvalue we de ne the so callednorming constants and relate them to the spectral measure and the m matrix thisenables us to recover from the m matrix the boundary conditions and the potential up to a unitary equivalence for co normal boundary conditions keywords inverse problem di erential operators on graphs m function sturm liouville 2000 msc 34a55 34b45 34b20 34l05 34b27

a front tracking method for the computations of multiphase flow direct numerical simulations of multiphase flows using a front tracking method are presented the method is based on writing one set of governing equations for the whole computational domain and treating the different phases as one fluid with variable material properties interfacial terms are accounted for by adding the appropriate sources as functions at the boundary separating the phases the unsteady navier stokes equations are solved by a conventional finite volume method on a fixed structured grid and the interface or front is tracked explicitly by connected marker points interfacial source terms such as surface tension are computed on the front and transferred to the fixed grid advection of fluid properties such as density is done by following the motion of the front the method has been implemented for fully three dimensional flows as well as for two dimensional and axisymmetric ones first the method is described for the flow of two or more isothermal phases the representation of the moving interface and its dynamic restructuring as well as the transfer of information between the moving front and the fixed grid are discussed applications and extensions of the method to homogeneous bubbly flows atomization flows with variable surface tension solidification and boiling are then presented

existence of weak solutions for the motion of rigid bodies in a viscous fluid we study the evolution of a finite number of rigid bodies within a viscous incompressible fluid in a bounded domain of r d d 2 or 3 with dirichlet boundary conditions by introducing an appropriate weak formulation for the complete problem we prove existence of solutions for initial velocities in h 1 0 omega in the absence of collisions solutions exist for all time in dimension 2 whereas in dimension 3 the lifespan of solutions is infinite only for small enough data

interfacial dynamics for stokes flow theoretical and computational aspects of the method of interfacial dynamics for flow at vanishing reynolds number are considered the mathematical formulation relies on the boundary integral representation that expresses the flow in terms of distributions of stokes flow singularities over the interfaces the densities of the distributions are identified with the jump in the hydrodynamic traction due to interfacial in plane and transverse tensions the interfacial velocity or the strength of a hydrodynamic potential the numerical procedure involves describing the interfaces in terms of interfacial marker points that reproduce the evolving shapes of the interfaces by global or local interpolation solving integral equations of the second kind for the interfacial velocity or for the density of a hydrodynamic potential and computing the motion of the marker points while simultaneously updating interfacial fields relevant to the dynamics including the concentration of a surfactant and the position of interfacial point particles at an equilibrium configuration interfaces exhibiting isotropic tension elastic tensions viscous and incompressible behavior are considered the mathematical modeling of the tensions and bending moments developing over interfaces with a membrane like constitution is discussed in the context of the theory of thin shells to facilitate the numerical implementation the coupling of the interfacial mechanics to the hydrodynamics by means of interface force and torque balances is formulated in global cartesian coordinates recent progress in the implementation of boundary element methods is reviewed and areas for further research are identified