an approximation to δ couplings on graphs we discuss a general parametrization for vertices of quantum graphs and show in particular how the s and couplings at an n edge vertex can be approximated by means of n 1 couplings of the type provided the latter are properly scaled

approximation of quantum graph vertex couplings by scaled schrodinger operators on thin branched manifolds we discuss approximations of vertex couplings of quantum graphs using families of thin branched manifolds we show that if a neumann type laplacian on such manifolds is amended by suitable potentials the resulting schrodinger operators can approximate non trivial vertex couplings the latter include not only the couplings but also those with wavefunctions discontinuous at the vertex we work out the example of the symmetric couplings and make a conjecture that the same method can be applied to all couplings invariant with respect to the time reversal we conclude with a result that certain vertex couplings cannot be approximated by a pure laplacian

a posteriori error estimates for webster s equation in wave propagation we consider a generalised webster s equation for describing wave propagation in curved tubular structures such as variable diameter acoustic wave guides webster s equation in generalised form has been rigorously derived in a previous article starting from the wave equation and it approximates cross sectional averages of the propagating wave here the approximation error is estimated by an a posteriori technique

a finite element framework for continua with boundary energies part ii the three dimensional case this paper in line with part i 53 is concerned with the finite element implementation of boundary potential energies and the study of their impact on the deformations of solids thereby the main thrust is the fully three dimensional formulation and implementation incorporating anisotropic effects boundary effects can play a dominant role in the material behavior the most prominent example being surface tension however the common modelling in continuum mechanics takes exclusively the bulk into account nevertheless neglecting possible contributions from the boundary within this contribution the boundary potentials are allowed in general to depend not only on the boundary deformation but also on the boundary deformation gradient and the spatial boundary normal for the formulation of the finite element method the concept of convected curvilinear coordinates attached to the boundary is employed and the corresponding derivations completely based on a tensorial representation are carried out afterwards the discretization of the generalized weak formulation including boundary potentials is performed and eventually numerical examples are presented to demonstrate the boundary effects due to different proposed material models in contrast to the previous literature on this topic the current manuscript covers jointly the following issues related to boundary energies 1 the formulation and implementation represents a fully three dimensional framework at large deformations 2 the formulation of the problem and the proposed material models are based on finite strains however it is shown that the linearization would lead to the small strain models proposed previously in the literature and 3 the current manuscript covers the issue of anisotropy effects on the boundary energies which to the best of our knowledge has not been exemplified earlier

spectra of graph neighborhoods and scattering let g e e 0 be a family of e thin riemannian manifolds modeled on a finite metric graph g for example the e neighborhood of an embedding of g in some euclidean space with straight edges we study the asymptotic behavior of the spectrum of the laplace beltrami operator on g e as e 0 for various boundary conditions we obtain complete asymptotic expansions for the kth eigenvalue and the eigenfunctions uniformly for k ce 1 in terms of scattering data on a non compact limit space we then use this to determine the quantum graph which is to be regarded as the limit object in a spectral sense of the family g e our method is a direct construction of approximate eigenfunctions from the scattering and graph data and the use of a priori estimates to show that all eigenfunctions are obtained in this way

a tool for the analysis of quasi newton methods with application to unconstrained minimization the bfgs update formula is shown to have an important property that is inde pendent of the algorithmic context of the update and that is relevant to both constrained and unconstrained optimization the bfgs method for unconstrained optimization using a variety of line searches including backtracking is shown to be globally and superlinearly convergent on uniformly convex problems the analysis is particularly simple due to the use of some new tools introduced in this paper

global convergence of a cass of quasi newton methods on convex problems we study the global convergence properties of the restricted broyden class of quasi newton methods when applied to a convex objective function we assume that the line search satisfies a standard sufficient decrease condition and that the initial hessian approximation is any positive definite matrix we show global and superlinear convergence for this class of methods except for dfp this generalizes powell s well known result for the bfgs method the analysis gives us insight into the properties of these algorithms in particular it shows that dfp lacks a very desirable self correcting property possessed by bfgs

local convergence of a two piece update of a projected hessian matrix the problem considered is that of minimizing a nonlinear function subject to a set of equality constraints recent research has focused on methods that solve such problems by recurring an approximation to second derivative information projected onto the tangent space of the constraints the motivation for these methods is that only the projected matrix enters into the optimality conditions for the nonlinear problem and that the reduced matrix can be recurred by means of updates that maintain hereditary positive definiteness however such methods can achieve at best local two step q superlinear convergence the algorithm presented here performs a two piece update of a one sided projected hessian matrix this method maintains one piece of the projected hessian as a positive definite matrix and uses broyden s method for updating the other piece it is shown that if at least one of the updates is performed at each iteration the method is locally one step q superlinearly convergent

variational problems on multiply connected thin strips i basic estimates and convergence of the laplacian spectrum let m be a planar embedded graph whose arcs meet transversally at the vertices let be a strip shaped domain around m of width except in a neighborhood of the singular points assume that the boundary of is smooth we define comparison operators between functions on and on m and we derive energy estimates for the compared functions we define a laplace operator on m which is in a certain sense the limit of the laplace operator on with neumann boundary conditions in particular we show that the p th eigenvalue of the laplacian on converges to the p th eigenvalue of the laplacian on m as tends to 0 a similar result holds for the magnetic schrodinger operator

coupling in the singular limit of thin quantum waveguides we analyze the problem of approximating a smooth quantum waveguide with a quantum graph we consider a planar curve with compactly supported curvature and a strip of constant width around the curve we rescale the curvature and the width in such a way that the strip can be approximated by a singular limit curve consisting of one vertex and two infinite straight edges i e a broken line we discuss the convergence of the laplacian with dirichlet boundary conditions on the strip in a suitable sense and we obtain two possible limits the laplacian on the line with dirichlet boundary conditions in the origin and a nontrivial family of point perturbations of the laplacian on the line the first case generically occurs and corresponds to the decoupling of the two components of the limit curve while in the second case a coupling takes place we present also two families of curves which give rise to coupling

zeta functions of quantum graphs in this paper we construct zeta functions of quantum graphs using a contour integral technique based on the argument principle we start by considering the special case of the star graph with neumann matching conditions at the center of the star we then extend the technique to allow any matching conditions at the center for which the laplace operator is self adjoint and finally obtain an expression for the zeta function of any graph with general vertex matching conditions in the process it is convenient to work with new forms for the secular equation of a quantum graph that extend the well known secular equation of the neumann star graph in the second half of this paper we apply the zeta function to obtain new results for the spectral determinant vacuum energy and heat kernel coefficients of quantum graphs these have all been topics of current research in their own right and in each case this unified approach significantly expands results in the literature

the berry keating operator on l 2 mathbb r _ gt rm d x and on compact quantum graphs with general self adjoint realizations the berry keating operator berry and keating 1999 siam rev 41 236 governing the schrodinger dynamics is discussed in the hilbert space and on compact quantum graphs it has been proved that the spectrum of hbk defined on is purely continuous and thus this quantization of hbk cannot yield the hypothetical hilbert polya operator possessing as eigenvalues the nontrivial zeros of the riemann zeta function a complete classification of all self adjoint extensions of hbk acting on compact quantum graphs is given together with the corresponding secular equation in form of a determinant whose zeros determine the discrete spectrum of hbk in addition an exact trace formula and the weyl asymptotics of the eigenvalue counting function are derived furthermore we introduce the squared berry keating operator which is a special case of the black scholes operator used in financial theory of option pricing again all self adjoint extensions the corresponding secular equation the trace formula and the weyl asymptotics are derived for h2bk on compact quantum graphs while the spectra of both hbk and h2bk on any compact quantum graph are discrete their weyl asymptotics demonstrate that neither hbk nor h2bk can yield as eigenvalues the nontrivial riemann zeros some simple examples are worked out in detail

the generalized star product and the factorization of scattering matrices on graphs in this article we continue our analysis of schrodinger operators on arbitrary graphs given as certain laplace operators in the present article we give the proof of the composition rule for the scattering matrices this composition rule gives the scattering matrix of a graph as a generalized star product of the scattering matrices corresponding to its subgraphs we perform a detailed analysis of the generalized star product for arbitrary unitary matrices the relation to the theory of transfer matrices is also discussed

branched quantum wave guides with dirichlet boundary conditions the decoupling case we consider a family of open sets m which shrinks with respect to an appropriate parameter to a graph under the additional assumption that the vertex neighbourhoods are small we show that the appropriately shifted dirichlet spectrum of m converges to the spectrum of the differential laplacian on the graph with dirichlet boundary conditions at the vertices i e a graph operator without coupling between different edges the smallness is expressed by a lower bound on the first eigenvalue of a mixed eigenvalue problem on the vertex neighbourhood the lower bound is given by the first transversal mode of the edge neighbourhood we also allow curved edges and show that all bounded eigenvalues converge to the spectrum of a laplacian acting on the edge with an additional potential coming from the curvature

convergence of spectra of uniformly fattened open book structures abstract we consider a stratified compact surface m in r 3 which we call an open book structure whose singular points form simple closed smooth curves and its neighborhood m which we call a fattened open book structure assuming a transversality condition at the singularities and absence of zero dimensional strata i e corners we show that the discrete spectrum of the neumann laplacian in m converges when 0 to the spectrum of a differential operator on m similar results have been obtained before for the case of fattened graphs i e m being one dimensional in the case of a smooth surfaces m the problem has been studied well however having singularities along strata of lower dimensions significantly complicates considerations as in the quantum graph case such considerations are triggered by various applications

ale method for solidification modelling abstract in this paper an arbitrary lagrangian eulerian ale formulation has been developed for the simulation of casting processes the method is applied both to mould filling simulation where it can provide accurate free surface description and to the study of thermo mechanical phenomena occurring in the subsequent cooling down of cast parts prediction of solute transport and of distortions and stresses in the first three sections all governing equations constitutive equations momentum energy solute transport are given then the ale formulation is presented mesh updating with evolving free surfaces using conservative normal vectors to enforce mass conservation treatment of advection terms using an original nodal upwind method definition of lagrangian and eulerian lagrangian regions finally examples of applications are given

norm resolvent convergence of one dimensional high contrast periodic problems to a kronig penney dipole type model we prove operator norm resolvent convergence estimates for one dimensional periodic differential operators with rapidly oscillating coefficients in the non uniformly elliptic high contrast setting which has been out of reach of the existing homogenisation techniques our asymptotic analysis is based on a special representation of the resolvent of the operator in terms of the m matrix of an associated boundary triple krein resolvent formula the resulting asymptotic behaviour is shown to be described up to a unitary transformation by a non standard version of the kronig penney model on mathbb r

an analysis of reduced hessian methods for constrained optimization we study the convergence properties of reduced hessian successive quadratic programming for equality constrained optimization the method uses a backtracking line search and updates an approximation to the reduced hessian of the lagrangian by means of the bfgs formula two merit functions are considered for the line search thel1 function and the fletcher exact penalty function we give conditions under which local and superlinear convergence is obtained and also prove a global convergence result the analysis allows the initial reduced hessian approximation to be any positive definite matrix and does not assume that the iterates converge or that the matrices are bounded the effects of a second order correction step a watchdog procedure and of the choice of null space basis are considered this work can be seen as an extension to reduced hessian methods of the well known results of powell 1976 for unconstrained optimization

characteristic functions of quantum graphs in this paper we give a reduction formula for the characteristic functions of the sturm liouville boundary value problems defined on a tree we also discuss the multiplicity of the eigenvalues and interlacing properties between two spectral sets associated with different boundary conditions

imaging geometric graphs using internal measurements abstract this article presents two topologically motivated algorithms for reconstructing a metric graph from solutions to a wave equation on it our algorithms rely on narrowband and visibility measurements and are therefore of considerable value to urban sensing applications this is an advancement over the wideband methods traditionally used to study metric graphs our methodology exposes and separates the impact of the graph topology and geometry and is driven by a complete cohomological characterization of the space of wave equation solutions

partitioned quasi newton methods for nonlinear equality constrained optimization we examine the nonlinear equality constrained problem nep in a projected or partitioned form to reveal the basic structure of the hessian matrix the concepts of quasi newton methods are then introduced by considering unconstrained optimization problems drawing on this background we present quasi newton methods for nep and explore properties of the approximate hessian matrix which insure good performance of sequential quadratic programming sqp type algorithms positivity quasi newton relation symmetry and the least change property current quasi newton methods and the strategies they use to provide these properties are then examined although it is particularly desirable to maintain positivity of the null space projection of the hessian matrix we show that existing methods fail to do this in a simple and reliable manner consequently our goal in this thesis is to find new updates which incorporate these desirable properties of the hessian matrix using a variational approach we derive several new quasi newton updates for nep these updates are derived using the partitioned forms of the hessian matrix allowing us to incorporate the desired properties for each partition two of the update formulas the lower half update lhu and the partitioned bfgs update pbu are shown to be particularly successful the lhu and pbu formulas are of the least change type and satisfy a standard quasi newton equation retain symmetry of the full hessian and keep the null space projection of the hessian positive definite in special cases these new updates simplify to projected forms of the broyden powell symmetric broyden and broyden fletcher goldfarb shanno formulas these new updates are particularly attractive since they require only one gradient evaluation per iteration they may be used at nearly every iteration even when we are far from the solution without requiring special line search conditions or step size restrictions next we discuss implementation of the lhu and pbu formulas in particular we examine the cubic equation which must be solved at every iteration its accurate solution and the selection of the cubic root we prove r superlinear convergence and present extensive computational results including comparisons with existing quasi newton methods for the nonlinear equality problem

parallel lagrange newton krylov schur methods for pde constrained optimization part i the krylov schur solver large scale optimization of systems governed by partial differential equations pdes is a frontier problem in scientific computation reduced quasi newton sequential quadratic programming sqp methods are state of the art approaches for such problems these methods take full advantage of existing pde solver technology and parallelize well however their algorithmic scalability is questionable for certain problem classes they can be very slow to converge in this two part article we propose a new method for steady state pde constrained optimization based on the idea of using a full space newton solver combined with an approximate reduced space quasi newton sqp preconditioner the basic components of the method are newton solution of the first order optimality conditions that characterize stationarity of the lagrangian function krylov solution of the karush kuhn tucker kkt linear systems arising at each newton iteration using a symmetric quasi minimum residual method preconditioning of the kkt system using an approximate state decision variable decomposition that replaces the forward pde jacobians by their own preconditioners and the decision space schur complement the reduced hessian by a bfgs approximation initialized by a two step stationary method accordingly we term the new method it lagrange newton krylov schur lnks it is fully parallelizable exploits the structure of available parallel algorithms for the pde forward problem and is locally quadratically convergent in part i of this two part article we investigate the effectiveness of the kkt linear system solver we test our method on two optimal control problems in which the state constraints are described by the steady state stokes equations the objective is to minimize dissipation or the deviation from a given velocity field the control variables are the boundary velocities numerical experiments on up to 256 cray t3e processors and on an sgi origin 2000 include scalability and performance assessment of the lnks algorithm and comparisons with reduced sqp for up to 1 000 000 state and 50 000 decision variables in part ii of the article we address globalization and inexactness issues and apply lnks to the optimal control of the steady incompressible navier stokes equations

a single mode quantum transport in serial structure geometric scatterers we study transport in quantum systems consisting of a finite array of n identical single channel scatterers a general expression of the s matrix in terms of the individual element data obtained recently for potential scattering is rederived in this wider context it shows in particular how the band spectrum of the infinite periodic system arises in the limit n we illustrate the result on two kinds of examples the first are serial graphs obtained by chaining loops or t junctions another example concerns geometric scatterers where the individual element consists of a surface with a pair of leads we show that apart from the resonances coming from the decoupled surface eigenvalues such scatterers exhibit the high energy behavior typical for the interaction for the physically interesting couplings

index theorems for quantum graphs in geometric analysis an index theorem relates the difference of the numbers of solutions of two differential equations to the topological structure of the manifold or bundle concerned sometimes using the heat kernels of two higher order differential operators as an intermediary in this paper the case of quantum graphs is addressed a quantum graph is a graph considered as a singular one dimensional variety and equipped with a second order differential hamiltonian h a laplacian with suitable conditions at vertices for the case of scale invariant vertex conditions i e conditions that do not mix the values of functions and of their derivatives the constant term of the heat kernel expansion is shown to be proportional to the trace of the internal scattering matrix of the graph this observation is placed into the index theory context by factoring the laplacian into two first order operators h a a and relating the constant term to the index of a an independent consideration provides an index formula for any differential operator on a finite quantum graph in terms of the vertex conditions it is also found that the algebraic multiplicity of 0 as a root of the secular determinant of h is the sum of the nullities of a and a

an arbitrary lagrangian eulerian σ ales model with non hydrostatic pressure for shallow water flows abstract an arbitrary lagrangian eulerian model in the coordinate system ales is developed for shallow water flows based on the unsteady reynolds averaged navier stokes equations unlike the conventional coordinate system non hydrostatic pressure is incorporated and the effect of a moving free surface included giving a general scheme the standard k turbulence model is used to calculate the eddy viscosity wave flows over bars are appropriate test cases for which experimental data are available and comparisons are favourable the model is also applied to the important practical problem of wave and wave current flows over a trench for which experimental data are available only for the steady current alone case the inclusion of vertical grid velocity has negligible effect on computational efficiency the solution of poisson s equation for non hydrostatic pressure at each time step is solved efficiently by the conjugate gradient method but represents the main component of computer time the model may readily be extended to three dimensions

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

inverse eigenvalue problems on directed graphs the differential operators id and d2 p are constructed on certain finite directed weighted graphs two types of inverse spectral problems are considered first information about the graph weights and boundary conditions is extracted from the spectrum of d2 second the compactness of isospectral sets for d2 p is established by computation of the residues of the zeta function

spectral properties of matrices associated with some directed graphs we study the spectral properties of certain non self adjoint matrices associated with large directed graphs asymptotically the eigenvalues converge to certain curves apart from a finite number that have limits not on these curves

quantum graphs as holonomic constraints we consider the dynamics on a quantum graph as the limit of the dynamics generated by a one particle hamiltonian in r2 with a potential having a deep strict minimum on the graph when the width of the well shrinks to zero for a generic graph we prove convergence outside the vertices to the free dynamics on the edges for a simple model of a graph with two edges and one vertex we prove convergence of the dynamics to the one generated by the laplacian with dirichlet boundary conditions in the vertex

an example of irregular convergence in some constrained optimization methods that use the projected hessian in this paper we give examples illustrating the behavior of the coleman conn horizontal vertical method and of successive quadratic programming with a hessian approximation exact on the tangent space of the constraints one example shows that these methods in general are not one step superlinearly convergent

the spectrum of the hilbert space valued second derivative with general self adjoint boundary conditions abstract we consider a large class of self adjoint elliptic problems associated with the second derivative acting on a space of vector valued functions we present and survey several results that can be obtained by means of two different approaches to the study of the associated eigenvalues problems the first more general one allows to replace a secular equation which is well known in some special cases by an abstract rank condition the second one though available in general seems to apply particularly well to a specific boundary condition the sometimes dubbed anti kirchhoff condition in the literature that arises in the theory of differential operators on graphs it also permits to discuss interesting and more direct connections between the spectrum of the differential operator and some graph theoretical quantities in particular some results on the symmetry of the spectrum in either case

eigenfunctions of the edge based laplacian on a graph abstract in this paper we analyze the eigenfunctions of the edge based laplacian on a graph and the relationship of these functions to random walks on the graph we commence by discussing the set of eigenfunctions supported at the vertices and demonstrate the relationship of these eigenfunctions to the classical random walk on the graph then from an analysis of functions supported only on the interior of edges we develop a method for explicitly calculating the edge interior eigenfunctions of the edge based laplacian this reveals a connection between the edge based laplacian and the adjacency matrix of backtrackless random walk on the graph the edge based eigenfunctions therefore correspond to some eigenfunctions of the normalized hashimoto matrix

an only 2 step q superlinear convergence example for some algorithms that use reduced hessian approximations it is shown by example that the reduced hessian method for constrained optimization that is known to give 2 stepq superlinear convergence may not convergeq superlinearly

error estimates and convergence rates for various incompressible elements this note reports numerical experiments on the efficiency of simple error estimates derived earlier1 applied to incompressible mixed or related penalty type formulations the rate of convergence and performance of various mixed elements is compared numerical results from a driven cavity and an incompressible elastic problem demonstrate that the t6b1 3d and t6 3c elements give a faster rate of convergence than the t6 1d element however in the case of a plane extrusion analysis stronger singularity the rate of convergence for the t6b1 3d element drops and is inferior to that of the t6 1d while the t6 3c element still proves superior to the other two elements

arbitrary lagrangian eulerian finite element analysis of free surface flow using a velocity vorticity formulation this paper describes the application of velocity vorticity formulation of the navier stokes equations for two dimensional free surface flow using an arbitrary lagrangian eulerian method the velocity poisson equations and the vorticity transport equations are solved using a finite element method to obtain the velocity and the vorticity fields in the interior region of the computational domain the boundary fitted coordinates system is adopted to solve the boundary equations for kinematic and dynamic conditions at the free surface using a finite difference method the numerical model for the velocity vorticity formulation is validated for a square cavity flow at re 400 and 1000 the solitary wave reflected from a vertical wall is chosen as a test case for comparison and validation of the free surface flow model then the proposed numerical model is used to obtain flow results for the following free surface flow cases i interaction between two opposite solitary waves ii seiche phenomenon in a rectangular reservoir and iii solitary wave through a submerged rectangular structure in a viscous fluid the efficiency of the present numerical model for numerical treatment of free surface flows is discussed furthermore the advantage of this formulation with respect to primitive variables formulation is addressed from the computational point of view

a fictitious domain approach for a mixed finite element method solving the two phase stokes problem with surface tension forces abstract in this article we study a mixed finite element formulation for solving the stokes problem with general surface forces that induce a jump of the normal trace of the stress tensor on an interface that splits the domain into two subdomains equality of velocities is assumed at the interface the interface conditions are taken into account with multipliers a suitable lagrangian functional is introduced for deriving a weak formulation of the problem a specificity of this work is the consideration of the interface with a fictitious domain approach the latter is inspired by the xfem approach in the sense that cut off functions are used but it is simpler to implement since no enrichment by singular functions is provided in that context getting convergence for the dual variables defined on the interface is non trivial for that purpose an augmented lagrangian technique stabilizes the convergence of the multipliers which is important because their value would determine the dynamics of the interface in an unsteady framework theoretical analysis is provided where we show that a discrete inf sup condition independent of the mesh size is satisfied for the stabilized formulation this guarantees optimal convergence rates that we observe with numerical tests the capacity of the method is demonstrated with robustness tests and with an unsteady model tested for deformations of the interface that correspond to ellipsoidal shapes in dimension 2

a finite element framework for continua with boundary energies part iii the thermomechanical case abstract this paper in line with the previous works javili and steinmann 28 29 is concerned with the numerical implementation of boundary potential energies and the study of their impact on the deformations of solids thereby the main thrust in this contribution is the extension to thermomechanical effects although boundary effects can play a dominant role in material behavior the common modelling in continuum mechanics takes exclusively the bulk into account nevertheless neglecting possible contributions from the boundary in the approach of this contribution the boundary is equipped with its own thermodynamic life i e we assume separate boundary energy entropy and the like furthermore the generalized local balance laws are given according to javili and steinmann 30 afterwards the derivations of a generalized weak formulation including boundary potentials completely based on a tensorial representation are carried out which is employed for the discretization and finite element implementation eventually numerical examples are presented to demonstrate the boundary effects due to the proposed thermohyperelastic material model

working range optimization for turbine and compressor blading a partially reduced sqp algorithm for a multiple setpoint industrial design optimization problem is presented it is highlighted that the modularity of this concept has made it possible to apply it to problems as complex as working range optimization in high temperature gas turbine and compressor blade design further emphasis is drawn on parallelization aspects and practical experiences in engineering practice

on the inverse scattering problem on branching graphs the inverse scattering problem on branching graphs is studied the definition of the schrodinger operator on such graphs is discussed the operator is defined with real potentials with finite first momentum and using special boundary conditions connecting values of the functions at the vertices it is shown that in general the scattering matrix does not determine the topology of the graph the potentials on the edges and the boundary conditions uniquely

some inverse scattering problems on star shaped graphs having in mind applications to the fault detection diagnosis of lossless electrical networks here we consider some inverse scattering problems for schrodinger operators over star shaped graphs we restrict ourselves to the case of minimal experimental setup consisting in measuring at most two reflection coefficients when an infinite homogeneous potential less branch is added to the central node first by studying the asymptotic behavior of only one reflection coefficient in the high frequency limit we prove the identiability of the geometry of this star shaped graph the number of edges and their lengths next we study the potential identification problem by inverse scattering noting that the potentials represent the inhomogeneities due to the soft faults in the network wirings potentials with bounded h1 norms the main result states that under some assumptions on the geometry of the graph the measurement of two reflection coefficients associated to two different sets of boundary conditions at the external vertices of the tree determines uniquely the potentials it can be seen as a generalization of the theorem of the two boundary spectra on an interval

experimental simulation of quantum graphs by microwave networks we present the results of experimental and theoretical study of irregular tetrahedral microwave networks consisting of coaxial cables annular waveguides connected by t joints the spectra of the networks were measured in the frequency range 0 0001 16 ghz in order to obtain their statistical properties such as the integrated nearest neighbor spacing distribution and the spectral rigidity d3sld the comparison of our experimental and theoretical results shows that microwave networks can simulate quantum graphs with time reversal symmetry in particular we use the spectra of the microwave networks to study the periodic orbits of the simulated quantum graphs we also present experimental study of directional microwave networks consisting of coaxial cables and faraday isolators for which the time reversal symmetry is broken in this case our experimental results indicate that spectral statistics of directional microwave networks deviate from predictions of gaussian orthogonal ensembles in random matrix theory approaching especially for small eigenfrequency spacing s results for gaussian unitary ensembles experimental results are supported by the theoretical analysis of directional graphs

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

nodal domains on isospectral quantum graphs the resolution of isospectrality we present and discuss isospectral quantum graphs which are not isometric these graphs are the analogues of the isospectral domains in which were introduced recently in gordon et al 1992 bull am math soc 27 134 8 chapman 1995 am math mon 102 124 buser et al 1994 int math res not 9 391 400 okada and shudo 2001 j phys a math gen 34 5911 22 jakobson et al 2006 j comput appl math 194 141 55 and levitin et al 2006 j phys a math gen 39 2073 82 all based on sunada s construction of isospectral domains sunada t 1985 ann math 121 196 86 after presenting some of the properties of these graphs we discuss a few examples which support the conjecture that by counting the nodal domains of the corresponding eigenfunctions one can resolve the isospectral ambiguity

optimal control of two and three dimensional incompressible navier stokes flows the focus of this work is on the development of large scale numerical optimization methods for optimal control of steady incompressible navier stokes flows the control is affected by the suction or injection of fluid on portions of the boundary and the objective function represents the rate at which energy is dissipated in the fluid we develop reduced hessian sequential quadratic programming methods that avoid converging the flow equations at each iteration both quasi newton and newton variants are developed and compared to the approach of eliminating the flow equations and variables which is effectively the generalized reduced gradient method optimal control problems are solved for two dimensional flow around a cylinder and three dimensional flow around a sphere the examples demonstrate at least an order of magnitude reduction in time taken allowing the optimal solution of flow control problems in as little as half an hour on a desktop workstation

a topological state sum model for fermions on the circle a simple state sum model for fermions on a 1 manifold is constructed the model is independent of the triangulation and gives exactly the same partition function as the dirac functional integral with zeta function regularization some implications for more realistic physical models are discussed

asymptotically isospectral quantum graphs and generalised trigonometric polynomials abstract the theory of almost periodic functions is used to investigate spectral properties of schrodinger operators on metric graphs also known as quantum graphs in particular we prove that two schrodinger operators may have asymptotically close spectra if and only if the corresponding reference laplacians are isospectral our result implies that a schrodinger operator is isospectral to the standard laplacian on a may be different metric graph only if the potential is identically equal to zero

on the convergence of constrained optimization methods with accurate hessian information on a subspace this paper analyzes methods of the type proposed by coleman and conn for nonlinearly constrained optimization it is shown that if the reduced hessian approximation is sufficiently accurate then the method generates a sequence of iterates that converges one step superlinearly this result applies to a quasi newton implementation if the reduced exact hessian is used the method has an r order equal to that of the secant method a similar result for a modified version of successive quadratic programming is also proved finally some parallels between convergence results for methods that approximate the reduced hessian method and multiplier methods that use the reduced hessian inverse are pointed out

infeasible path optimal design methods with applications to aerodynamic shape optimization we consider optimal design of physical systems described by a nonlinear boundary value problem we propose a sequential quadratic programming method specifically tailored to the solution of this class of design problems a particular representation of the null space of the constraint jacobian that exploits its specific structure is employed the resulting method avoids resolution of nonlinear behavior at the optimization iterations while keeping the size of the problem as small as that of conventional approaches three variants of the method are developed and discussed these entail the solution of either two or three linear systems involving the jacobian matrix of the discretized form of the boundary value problem the method is used to solve aerodynamic design problems involving nonlinear transonic flow no special provisions are made for treating discontinuities and therefore the present implementation of the method is limited to problems with no shocks problems with up to 90 shape design variables are solved numerical results demonstrate a substantial performance improvement over conventional methods

nonlinear programming via an exact penalty function global analysis in this paper we motivate and describe an algorithm to solve the nonlinear programming problem the method is based on an exact penalty function and possesses both global and superlinear convergence properties we establish the global qualities here the superlinear nature is proven in 7 the numerical implementation techniques are briefly discussed and preliminary numerical results are given

scattering on graphs and one dimensional approximations to n dimensional schrodinger operators in the present article we develop the spectral analysis of schrodinger operators on lattice type graphs for the basic example of a cubic periodic graph the problem is reduced to the spectral analysis of certain regular differential operators on a fundamental star like subgraph with a selfadjoint condition at the central node and quasiperiodic conditions at the boundary vertices using an explicit expression for the resolvent of lattice type operator we develop in the second section appropriate lippmann schwinger techniques for the perturbed periodic operator and construct the corresponding scattering matrix it serves as a base for the approximation of the multi dimensional schrodinger operator by a one dimensional operator on the graph in the third section of the paper for given n dimensional schrodinger operators with rapidly decreasing potential we construct a lattice type operator on a cubic graph embedded into rn and show that the original n dimensional scattering problem can be approximated in a p

a global convergence theory for dennis el alem and maciel s class of trust region algorithms for constrained optimization without assuming regularity this work presents a convergence theory for dennis el alem and maciel s class of trust region based algorithms for solving the smooth nonlinear programming problem with equality constraints the results are proved under very mild conditions on the quasi normal and tangential components of the trial steps the lagrange multiplier estimates and the hessian estimates are assumed to be bounded no regularity assumption is made in particular linear independence of the gradients of the constraints is not assumed the theory proves global convergence for the class in particular it shows that a subsequence of the iteration sequence satisfies one of four types of mayer bliss stationary conditions in the limit

a continuum method for modeling surface tension abstract a new method for modeling surface tension effects on fluid motion has been developed interfaces between fluids of different properties or colors are represented as transition regions of finite thickness across which the color variable varies continuously at each point in the transition region a force density is defined which is proportional to the curvature of the surface of constant color at that point it is normalized so that the conventional description of surface tension on an interface is recovered when the ratio of local transition region thickness to local radius of curvature approaches zero the continuum method eliminates the need for interface reconstruction simplifies the calculation of surface tension enables accurate modeling of two and three dimensional fluid flows driven by surface forces and does not impose any modeling restrictions on the number complexity or dynamic evolution of fluid interfaces having surface tension computational results for two dimensional flows are given to illustrate the properties of the method

an sqp method for the optimal control of large scale dynamical systems we propose a sequential quadratic programming sqp method for the optimal control of large scale dynamical systems the method uses modified multiple shooting to discretize the dynamical constraints when these systems have relatively few parameters the computational complexity of the modified method is much less than that of standard multiple shooting moreover the proposed method is demonstrably more robust than single shooting in the context of the sqp method the use of modified multiple shooting involves a transformation of the constraint jacobian the affected rows are those associated with the continuity constraints and any path constraints applied within the shooting intervals path constraints enforced at the shooting points and other constraints involving only discretized states are not transformed the transformation is cast almost entirely at the user level and requires minimal changes to the optimization software we show that the modified quadratic subproblem yields a descent direction for the l1 penalty function numerical experiments verify the efficiency of the modified method

scattering solutions in networks of thin fibers small diameter asymptotics small diameter asymptotics is obtained for scattering solutions in a network of thin fibers the asymptotics is expressed in terms of solutions of related problems on the limiting quantum graph we calculate the lagrangian gluing conditions at vertices v in gamma for the problems on the limiting graph if the frequency of the incident wave is above the bottom of the absolutely continuous spectrum the gluing conditions are formulated in terms of the scattering data for each individual junction of the network

a reduced hessian algorithm with line search filter method for nonlinear programming this paper proposes a line search filter reduced hessian method for nonlinear equality constrained optimization the feature of the presented algorithm is that the reduced hessian method is used to produce a search direction a backtracking line search procedure to generate step size some filtered rules to determine step acceptance second order correction technique to reduce infeasibility and overcome the maratos effects it is shown that this algorithm does not suffer from the maratos effects by using second order correction step and under mild assumptions fast convergence to second order sufficient local solutions is achieved the numerical experiment is reported to show the effectiveness of the proposed algorithm

convergence of spectra of mesoscopic systems collapsing onto a graph abstract let m be a finite graph in the plane and let me be a domain that looks like the e fattened graph m exact conditions on the domain are given it is shown that the spectrum of the neumann laplacian on me converges when e 0 to the spectrum of an ode problem on m the presence of an electromagnetic field is also allowed considerations of this kind arise naturally in mesoscopic physics and other areas of physics and chemistry the results of the paper extend the ones previously obtained by j rubinstein and m schatzman

finite element modelling of surface tension effects using a lagrangian eulerian kinematic description abstract an algorithm has been devised to model surface tension forces during fluid motion it makes use of an eulerian formulation applied to the governing equations and a lagrangian treatment of the free surface it is assumed that the effect of air on the free surface movement is negligible thus only the fluid domain is considered during the analysis the free surface is subjected to a surface tension force proportional to the local curvature this manifests itself as a simple and accurate method of modelling surface tension effects by virtue of the free surface being explicitly determined the method is validated by modelling the effect of surface tension on a static rod a further numerical example is used to illustrate the effect of surface tension on a non equilibrium square rod wall adhesion modelling is demonstrated by modelling the wetting of a capillary tube finally the method is used in an industrial context to model the filling of mould cavities for which surface tension forces are important

non weyl asymptotics for quantum graphs with general coupling conditions inspired by a recent result of davies and pushnitski we study resonance asymptotics of quantum graphs with general coupling conditions at the vertices we derive a criterion for the asymptotics to be of a non weyl character we show that for balanced vertices with permutation invariant couplings the asymptotics is non weyl only in the case of kirchhoff or anti kirchhoff conditions while for graphs without permutation symmetry numerous examples of non weyl behaviour can be constructed furthermore we present an insight into what makes the kirchhoff anti kirchhoff coupling particular from the resonance point of view finally we demonstrate a generalization to quantum graphs with unequal edge weights

diffusion processes in thin tubes and their limits on graphs the present paper is concerned with diffusion processes running on tubular domains with conditions on nonreaching the boundary respectively reflecting at the boundary and corresponding processes in the limit where the thin tubular domains are shrinking to graphs the methods we use are probabilistic ones for shrinking we use big potentials respectively reflection on the boundary of tubes we show that there exists a unique limit process and we characterize the limit process by a second order differential generator acting on functions defined on the limit graph with kirchhoff boundary conditions at the vertices

on occurrence of spectral edges for periodic operators inside the brillouin zone the paper discusses the following frequently arising question on the spectral structure of periodic operators of mathematical physics e g schrodinger maxwell waveguide operators etc is it true that one can obtain the correct spectrum by using the values of the quasimomentum running over the boundary of the reduced brillouin zone only rather than the whole zone or do the edges of the spectrum occur necessarily at the set of corner high symmetry points this is known to be true in 1d while no apparent reasons exist for this to be happening in higher dimensions in many practical cases though this appears to be correct which sometimes leads to the claims that this is always true there seems to be no definite answer in the literature and one encounters different opinions about this problem in the community in this paper starting with simple discrete graph operators we construct a variety of convincing multiply periodic examples showing that the spectral edges might occur deeply inside the brillouin zone on the other hand it is also shown that in a generic case the situation of spectral edges appearing at high symmetry points is stable under small perturbations this explains to some degree why in many maybe even most practical cases the statement still holds

a reduced hessian method for large scale constrained optimization we propose a quasi newton algorithm for solving large optimization problems with nonlinear equality constraints it is designed for problems with few degrees of freedom and is motivated by the need to use sparse matrix factorizations the algorithm incorporates a correction vector that approximates the cross term z t wy p y in order to estimate the curvature in both the range and null spaces of the constraints the algorithm can be considered to be in some sense a practical implementation of an algorithm of coleman and conn we give conditions under which local and superlinear convergence is obtained