parametric galois extensions abstract given a field k and a finite group h an h parametric extension over k is a finite galois extension of k t of galois group containing h which is regular over k and has all the galois extensions of k of group h among its specializations we are mainly interested in producing non h parametric extensions which relates to classical questions in inverse galois theory like the beckmann black problem and the existence of one parameter generic polynomials we develop a general approach started in a preceding paper and provide new non parametricity criteria and new examples

a sharp interface immersed boundary method for compressible viscous flows an immersed boundary method for computing viscous subsonic compressible flows with complex shaped stationary immersed boundaries is presented the method employs a ghost cell technique for imposing the boundary conditions on the immersed boundaries the current approach leads to a sharp representation of the immersed boundaries a property that is especially useful for flow simulations at high reynolds numbers another unique feature of the method is that it can be applied on cartesian as well as generalized body non conformal curvilinear meshes a mixed second order central difference quick scheme is used which allows a high degree of control over the numerical damping a bilinear interpolation scheme used in conjunction with the ghost cell approach results in second order global as well as local spatial accuracy the solver is parallelized for distributed memory platforms using domain decomposition and message passing interface mpi and salient features of the parallel algorithm are presented the accuracy fidelity and efficiency of the solver are examined by simulating flow past circular cylinders and airfoils and comparing against experimental data and other established results finally we present results from a simulation of wing tip flow at a relatively high reynolds number in order to demonstrate the ability of the solver to model complex non canonical three dimensional flows

degree of mobility for metrics of lorentzian signature and parallel 0 2 tensor fields on cone manifolds degree of mobility of a pseudo riemannian metric is the dimension of the space of metrics geodesically equivalent to it we describe all possible values of the degree of mobility on a simply connected n dimensional manifold of lorentzian signature as an application we calculate all possible differences between the dimension of the projective and the isometry groups one of the main new technical results in the proof is the description of all parallel symmetric 0 2 tensor fields on cone manifolds of signature n 1 2

balanced model reduction via the proper orthogonal decomposition a new method for performing a balanced reduction of a high order linear system is presented the technique combines the proper orthogonal decomposition and concepts from balanced realization theory the method of snapshotsisused to obtainlow rank reduced rangeapproximationsto thesystemcontrollability and observability grammiansineitherthetimeorfrequencydomain theapproximationsarethenusedtoobtainabalancedreducedorder model the method is particularly effective when a small number of outputs is of interest it is demonstrated for a linearized high order system that models unsteady motion of a two dimensional airfoil computation of the exact grammians would be impractical for such a large system for this problem very accurate reducedorder models are obtained that capture the required dynamics with just three states the new models exhibit far superiorperformancethanthosederived using a conventionalproperorthogonal decomposition although further development is necessary the concept also extends to nonlinear systems

a spectral viscosity method for correcting the long term behavior of pod models low dimensional flow dynamical systems may converge to erroneous states after long time integration even if they are initialized with the correct state in this paper we investigate the accuracy of such two dimensional models constructed from karhunen loeve expansions for flows past a circular cylinder we first demonstrate that although the short term dynamics may be predicted accurately with only a handful of modes retained drifting of the solution may arise after a few hundred vortex shedding cycles we then propose a dissipative model based on a spectral viscosity sv diffusion convolution operator the parameters of the sv model are selected rigorously based on bifurcation analysis our results show that this is an effective way of improving the accuracy of long term predictions of low dimensional galerkin systems

robust model identification of actuated vortex wakes we present a low order modeling technique for actuated flows based on the regularization of an inverse problem the inverse problem aims at minimizing the error between the model predictions and some reference simulations the parameters to be identified are a subset of the coefficients of a polynomial expansion which models the temporal dynamics of a small number of global modes these global modes are found by proper orthogonal decomposition which is a method to compute the most representative elements of an existing simulation database in terms of energy it is shown that low order control models based on a simple galerkin projection and usual calibration techniques are not viable they are either ill posed or they give a poor approximation of the solution as soon as they are used to predict cases not belonging to the original solution database in contrast numerical evidence shows that the method we propose is robust with respect to variations of the control laws applied thus allowing the actual use of such models for control

the variational multiscale method a paradigm for computational mechanics abstract we present a general treatment of the variational multiscale method in the context of an abstract dirichlet problem we show how the exact theory represents a paradigm for subgrid scale models and a posteriori error estimation we examine hierarchical p methods and bubbles in order to understand and ultimately approximate the fine scale green s function which appears in the theory we review relationships between residual free bubbles element green s functions and stabilized methods these suggest the applicability of the methodology to physically interesting problems in fluid mechanics acoustics and electromagnetics

a zonal galerkin free pod model for incompressible flows abstract a domain decomposition method which couples a high and a low fidelity model is proposed to reduce the computational cost of a flow simulation this approach requires to solve the high fidelity model in a small portion of the computational domain while the external field is described by a galerkin free proper orthogonal decomposition pod model we propose an error indicator to determine the extent of the interior domain and to perform an optimal coupling between the two models this zonal approach can be used to study multi body configurations or to perform detailed local analyses in the framework of shape optimisation problems the efficiency of the method to perform predictive low cost simulations is investigated for an unsteady flow and for an aerodynamic shape optimisation problem

hamiltonian 2 forms in kahler geometry ii global classification we present a classification of compact kahler manifolds admitting a hamiltonian 2 form which were classified locally in part i of this work this involves two components of independent interest the first is the notion of a rigid hamiltonian torus action this natural condition for torus actions on a kahler manifold was introduced locally in part i but such actions turn out to be remarkably well behaved globally leading to a fairly explicit classification up to a blow up compact kahler manifolds with a rigid hamiltonian torus action are bundles of toric kahler manifolds the second idea is a special case of toric geometry which we call orthotoric we prove that orthotoric kahler manifolds are diffeomorphic to complex projective space but we extend our analysis to orthotoric orbifolds where the geometry is much richer we thus obtain new examples of kahler einstein 4 orbifolds combining these two themes we prove that compact kahler manifolds with hamiltonian 2 forms are covered by blow downs of projective bundles over kahler products and we describe explicitly how the kahler metrics with a hamiltonian 2 form are parameterized we explain how this provides a context for constructing new examples of extremal kahler metrics in particular a subclass of such metrics which we call weakly bochner flat we also provide a self contained treatment of the theory of compact toric kahler manifolds since we need it and find the existing literature incomplete

a semi continuous formulation for goal oriented reduced order models 1d problems a semi continuous formulation is introduced for finding bases that minimise the error in a specific output functional of a reduced order model the formulation is advantageous in that it can be used with arbitrary reference data and can be easily applied to nonlinear models and functionals a general description of the approach is given then explicit formulations are derived for the advection diffusion and burgers equations numerical results are given for both linear and nonlinear functionals these show substantial reductions in error when compared with pod modes depending on the functional considered the optimisation of bases for a reduced order model using an approximated governing equation is also described for which large increases in accuracy are obtained relative to pod modes copyright c 2015 john wiley sons ltd

nonlinear model order reduction based on local reduced order bases summary a new approach for the dimensional reduction via projection of nonlinear computational models based on the concept of local reduced order bases is presented it is particularly suited for problems characterized by different physical regimes parameter variations or moving features such as discontinuities and fronts instead of approximating the solution of interest in a fixed lower dimensional subspace of global basis vectors the proposed model order reduction method approximates this solution in a lower dimensional subspace generated by most appropriate local basis vectors to this effect the solution space is partitioned into subregions and a local reduced order basis is constructed and assigned to each subregion offline during the incremental solution online of the reduced problem a local basis is chosen according to the subregion of the solution space where the current high dimensional solution lies this is achievable in real time because the computational complexity of the selection algorithm scales with the dimension of the lower dimensional solution space because it is also applicable to the process of hyper reduction the proposed method for nonlinear model order reduction is computationally efficient its potential for achieving large speedups while maintaining good accuracy is demonstrated for two nonlinear computational fluid and fluid structure electric interaction problems copyright 2012 john wiley sons ltd

a solution of a problem of sophus lie normal forms of two dimensional metrics admitting two projective vector fields we give a complete list of normal forms for the two dimensional metrics that admit a transitive lie pseudogroup of geodesic preserving transformations and we show that these normal forms are mutually non isometric this solves a problem posed by sophus lie

efficient non linear model reduction via a least squares petrov galerkin projection and compressive tensor approximations the first author acknowledges the partial support by a national science foundation graduate fellowship and the partial support by a national defense science and engineering graduate fellowship the second and third authors acknowledge the partial support by the motor sports division of the toyota motor corporation under agreement number 48737 and the partial support by a research grant from the academic excellence alliance program between king abdullah university of science and technology kaust and stanford university all authors also acknowledge the constructive comments received during the review process

tractor calculi for parabolic geometries parabolic geometries may be considered as curved analogues of the homogeneous spaces g p where g is a semisimple lie group and p c g a parabolic subgroup conformal geometries and cr geometries are examples of such structures we present a uniform description of a calculus called tractor calculus based on natural bundles with canonical linear connections for all parabolic geometries it is shown that from these bundles and connections one can recover the cartan bundle and the cartan connection in particular we characterize the normal cartan connection from this induced bundle connection perspective we construct explicitly a family of fundamental first order differential operators which are analogous to a covariant derivative iterable and defined on all natural vector bundles on parabolic geometries for an important subclass of parabolic geometries we explicitly and directly construct the tractor bundles their canonical linear connections and the machinery for explicitly calculating via the tractor calculus

optimal control of the cylinder wake in the laminar regime by trust region methods and pod reduced order models in this paper optimal control theory is used to minimize the total mean drag for a circular cylinder wake flow in the laminar regime re 200 the control parameters are the amplitude and the frequency of the time harmonic cylinder rotation in order to reduce the size of the discretized optimality system a proper orthogonal decomposition pod reduced order model rom is derived to be used as state equation we then propose to employ the trust region proper orthogonal decomposition trpod approach originally introduced by fahl m fahl trust region methods for flow control based on reduced order modeling ph d thesis trier university 2000 to update the reduced order models during the optimization process a lot of computational work is saved because the optimization process is now based only on low fidelity models a particular care was taken to derive a pod rom for the pressure and velocity fields with an appropriate balance between model accuracy and robustness the key enablers are the extension of the pod basis functions to the pressure data the use of calibration methods for the pod rom and the addition in the pod expansion of several non equilibrium modes to describe various operating conditions when the trpod algorithm is applied to the wake flow configuration this approach converges to the minimum predicted by an open loop control approach and leads to a relative mean drag reduction of 30 at reduced cost

supg reduced order models for convection dominated convection diffusion reaction equations abstract this paper presents a streamline upwind petrov galerkin supg reduced order model rom based on proper orthogonal decomposition pod this rom is investigated theoretically and numerically for convection dominated convection diffusion reaction problems the supg finite element method was used on realistic meshes for computing the snapshots leading to some noise in the pod data numerical analysis is used to propose the scaling of the stabilization parameter for the supg rom two approaches are used one based on the underlying finite element discretization and the other one based on the pod truncation the resulting supg roms and the standard galerkin rom g rom are studied numerically for many settings the results obtained with the supg roms are more accurate finally one of the choices for the stabilization parameter is recommended

α regularization of the pod galerkin dynamical systems of the kuramoto sivashinsky equation abstract in order to improve the dynamics and stability of the pod galerkin models of strongly stiff systems an like regularization is suggested and assessed in the present article in this method the pod eigenmodes of the non linear terms are replaced by their helmholtz filtered counterparts while the other terms are remained unchanged as an example the method is applied to the pod galerkin models of the one dimensional kuramoto sivashinsky ks equation in a full chaotic regime and the fidelity of the original and regularized models to the direct numerical simulations dns are investigated moreover the effects of regularization on the dynamics of various terms and whole of the systems are analyzed via eigenvalue analysis of each term separately and the total dynamical system as a whole the numerical experiments show definite effectiveness of the method and excellent improvements in the predicted dynamics and stability by minimum number of free parameters

on the use of pod based roms to analyze bifurcations in some dissipative systems abstract this paper deals with the use of pod based reduced order models to construct bifurcation diagrams which requires calculating steady and time dependent attractors in complex bifurcation problems involving dissipative systems the method proposed in the paper relies on the observation that pod manifolds resulting from snapshots calculated in time dependent runs for specific values of the parameters of the problem also contain the attractors for other values of the parameters the reason for this property is explained for a general class of dissipative systems which includes many problems of scientific industrial interest the consequence is that appropriate pod manifolds can be calculated in a quite computationally efficient way the method is illustrated considering both a simple bifurcation problem for a fisher like equation and a fairly complex bifurcation problem for the complex ginzburg landau equation

reduced order subscales for pod models in this work the reduced order subscales for proper orthogonal decomposition models are presented the basic idea consists in splitting the full order solution into the part which can be captured by the reduced order model and the part which cannot the subscales for which a model is required the proposed model for the subscales is defined as a linear function of the solution of the reduced order model the coefficients of this linear function are obtained by comparing the solution of the full order model with the solution of the reduced order model for the same initial conditions which for convenience are evaluated in the snapshots used to train the original reduced order model the difference between both solutions are the subscales for which a model can be built using a least squares procedure the subscales are then introduced as a correction in the reduced order model resulting in an important improvement in accuracy the enhanced reduced order model is tested in several numerical examples these practical cases show that the use of the subscales leads to more accurate solutions successfully corrects errors introduced by hyper reduction and allows to solve complex flow problems using a reduced number of degrees of freedom

geodesic hierarchies and involutivity we consider a special class of pseudo riemannian metrics that admit integrals in involution a necessary and sufficient conditions for the complete integrability of the geodesic flows of the corresponding metrics is proved any of the metrics we consider lies in a big family hierarchy of metrics that also admit integrals in involution many classical and new examples of completely integrable metrics lie in such hierarchies

central extensions of sn as galois groups of regular extensions of q t recently mestre me has proved that the double cover a of the alternating group a is realizable as the galois group of a regular extension of the rational function field uj t for all ii 3 4 we show here that his method yields the same result for the two double covers s of the symmetric group s z and more generally for every finite central extension of s partial results on s have been obtained previously by several authors kss sol so2 ss v

hamiltonian 2 forms in kahler geometry iii extremal metrics and stability this paper concerns the existence and explicit construction of extremal kahler metrics on total spaces of projective bundles which have been studied in many places we present a unified approach motivated by the theory of hamiltonian 2 forms as introduced and studied in previous papers in the series but this paper is largely independent of that theory

a nonreflecting outlet boundary condition for incompressible unsteady navier stokes calculations abstract the goal of this work is to adapt a nonreflecting outlet boundary condition derived from a wave equation to the numerical solution of the full incompressible navier stokes equations for an elliptic unsteady free shear flow the numerical results show that a significant improvement is achieved with this nonreflecting boundary condition in comparison with the results obtained by using free boundary layer type conditions the physical phenomena studied concern the onset of the kelvin helmholtz instability in the free non forced shear layer and certain 2d characteristics of transition towards turbulence these phenomena are simulated naturally without imposing perturbations the frequency of the organized vortices and the spread of the mixing layer are correctly predicted the performances of the method are shown through comparison with the physical experiments owing to the nonreflecting boundary conditions the feedback noises are inhibited effectively so that the computation domain can be reduced and the dynamic characteristics of the flow are maintained up clearly to the outlet boundary

metrisability of two dimensional projective structures we carry out the programme of r liouville 19 to construct an explicit local obstruction to the existence of a levi civita connection within a given projective structure on a surface the obstruction is of order 5 in the components of a connection in a projective class it can be expressed as a point invariant for a second order ode whose integral curves are the geodesics of or as a weighted scalar projective invariant of the projective class if the obstruction vanishes we find the sufficient conditions for the existence of a metric in the real analytic case in the generic case they are expressed by the vanishing of two invariants of order 6 in the connection in degenerate cases the sufficient obstruction isof order at most 8

pod deim reduced order strategies for efficient four dimensional variational data assimilation this work studies reduced order modeling rom approaches to speed up the solution of variational data assimilation problems with large scale nonlinear dynamical models it is shown that a key requirement for a successful reduced order solution is that reduced order karush kuhn tucker conditions accurately represent their full order counterparts in particular accurate reduced order approximations are needed for the forward and adjoint dynamical models as well as for the reduced gradient new strategies to construct reduced order based are developed for proper orthogonal decomposition pod rom data assimilation using both galerkin and petrov galerkin projections for the first time pod tensorial pod and discrete empirical interpolation method deim are employed to develop reduced data assimilation systems for a geophysical flow model namely the two dimensional shallow water equations numerical experiments confirm the theoretical framework for galerkin projection in the case of petrov galerkin projection stabilization strategies must be considered for the reduced order models the new reduced order shallow water data assimilation system provides analyses similar to those produced by the full resolution data assimilation system in one tenth of the computational time

on noether s problem for central extensions of symmetric and alternating groups for a field k and a finite group g acting regularly on a set of indeterminates x xg g g let k g denote the invariant field k x g we first prove for the alternating group an that if n is odd then q an is rational over q an 1 we then obtain an analogous result where an is replaced by an arbitrary finite central extension of either an or sn valid over q n for suitable n concrete applications of our results yield 1 a new proof of maeda s result on the rationality of q x1 x5 a5 q 2 an affirmative answer to noether s problem over q for both a5 and s5 3 an affirmative answer to noether s problem over c for every finite central extension group of either an or sn with n 5

some polynomials over q t and their galois groups examples of polynomials with galois group over q t corresponding to every transitive group through degree eight are calculated constructively demonstrating the existence of an infinity of extensions with each galois group over q through degree eight the methods used which for the most part have not appeared in print are briefly discussed

a spectral vanishing viscosity method for large eddy simulations a new simulation approach for high reynolds number turbulent flows is developed combining concepts of monotonicity in nonlinear conservation laws with concepts of large eddy simulation the spectral vanishing viscosity svv first introduced by e tadmor siam j numer anal 26 30 1989 is incorporated into the navier stokes equations for controlling high wavenumber oscillations unlike hyperviscosity kernels the svv approach involves a second order operator which can be readily implemented in standard finite element codes in the work presented here discretization is performed using hierarchical spectral hp methods accommodating effectively an ab initio intrinsic scale separation the key result is that monotonicity is enforced via svv leading to stable discretizations without sacrificing the formal accuracy i e exponential convergence in the proposed discretization several examples are presented to demonstrate the effectiveness of the new approach including a comparison with eddy viscosity spectral les of turbulent channel flow in its current implementation the svv approach for controlling the small scales is decoupled from the large scales but a procedure is proposed that will provide coupling similar to the classical les formulation

accelerating pde constrained optimization by the reduced basis method application to batch chromatography summary in this work we show that the reduced basis method accelerates a partial differential equation constrained optimization problem where a nonlinear discretized system with a large number of degrees of freedom must be repeatedly solved during optimization such an optimization problem arises for example from batch chromatography to reduce the computational burden of repeatedly solving the large scale system under parameter variations a parametric reduced order model with a small number of equations is derived by using the reduced basis method as a result the small reduced order model rather than the full system is solved at each step of the optimization process an adaptive technique for selecting the snapshots is proposed so that the complexity and runtime for generating the reduced basis are largely reduced an output oriented error bound is derived in the vector space whereby the construction of the reduced model is managed automatically an early stop criterion is proposed to circumvent the stagnation of the error and to make the construction of the reduced model more efficient numerical examples show that the adaptive technique is very efficient in reducing the offline time the optimization based on the reduced model is successful in terms of the accuracy and the runtime for acquiring the optimal solution copyright 2015 john wiley sons ltd

nonlinear model order reduction via lifting transformations and proper orthogonal decomposition this paper presents a structure exploiting nonlinear model reduction method for systems with general nonlinearities first the nonlinear model is lifted to a model with more structure via variable

splitting and gluing lemmas for geodesically equivalent pseudo riemannian metrics two metrics g and are geodesically equivalent if they share the same unparameterized geodesics we introduce two constructions that allow one to reduce many natural problems related to geodesically equivalent metrics such as the classification of local normal forms and the lie problem the description of projective vector fields to the case when the 1 1 tensor g g gkj has one real eigenvalue or two complex conjugate eigenvalues and give first applications as a part of the proof of the main result we generalise the topalov sinjukov hierarchy theorem for pseudo riemannian metrics 2011 american mathematical society

an analogue of serre s conjecture for galois representations and hecke eigenclasses in the mod p cohomology of gl n ℤ

a galerkin free model reduction approach for the navier stokes equations galerkin projection of the navier stokes equations on proper orthogonal decomposition pod basis is predominantly used for model reduction in fluid dynamics the robustness for changing operating conditions numerical stability in long term transient behavior and the pressure term consideration are generally the main concerns of the galerkin reduced order models rom in this article we present a novel procedure to construct an off reference solution state by using an interpolated pod reduced basis a linear interpolation of the pod reduced basis is performed by using two reference solution states the pod basis functions are optimal in capturing the averaged flow energy the energy dominant pod modes and corresponding base flow are interpolated according to the change in operating parameter the solution state is readily built without performing the galerkin projection of the navier stokes equations on the reduced pod space modes as well as the following time integration of the resulted ordinary differential equations ode to obtain the pod time coefficients the proposed interpolation based approach is thus immune from the numerical issues associated with a standard pod galerkin rom in addition a posteriori error estimate and a stability analysis of the obtained rom solution are formulated a detailed case study of the flow past a cylinder at low reynolds numbers is considered for the demonstration of proposed method the rom results show good agreement with the high fidelity numerical flow simulation

proof of the projective lichnerowicz conjecture for pseudo riemannian metrics with degree of mobility greater than two degree of mobility of a pseudo riemannian metric is the dimension of the space of metrics geodesically equivalent to it we prove that complete metrics on n 3 dimensional manifolds with degree of mobility 3 do not admit complete metrics that are geodesically equivalent to them but not affinely equivalent to them as the main application we prove an important special case of the pseudo riemannian version of the projective lichnerowicz conjecture stating that a complete manifold admitting an essential group of projective transformations is the standard round sphere up to a finite cover and multiplication of the metric by a constant

on the embedding problem for 2 s_4 representations let 2 s 4 denote the double cover of s 4 corresponding to the element in h 2 s 4 z 2z where transpositions lift to elements of order 2 and the product of two disjoint transpositions to elements of order 4 given an elliptic curve e let e 2 denote its 2 torsion points under some conditions on e elements in h 1 gal q e 2 0 correspond to galois extensions n of q with galois group isomorphic to s 4 in this work we give an interpretation of the addition law on such fields and prove that the obstruction for n having a galois extension n with gal n q 2 s 4 gives a homomorphism s 4 h 1 gal q e 2 h 2 gal q z 2z as a corollary we can prove if e has conductor divisible by few primes and high rank the existence of 2 dimensional representations of the absolute galois group of q attached to e and use them in some examples to construct 3 2 modular forms mapping via the shimura map to the modular form of weight 2 attached to e

model reduction for compressible flows using pod and galerkin projection abstract we present a framework for applying the method of proper orthogonal decomposition pod and galerkin projection to compressible fluids for incompressible flows only the kinematic variables are important and the techniques are well known in a compressible flow both the kinematic and thermodynamic variables are dynamically important and must be included in the configuration space we introduce an energy based inner product which may be used to obtain pod modes for this configuration space we then obtain an approximate version of the navier stokes equations valid for cold flows at moderate mach number and project these equations onto a pod basis the resulting equations of motion are quadratic and are much simpler than projections of the full compressible navier stokes equations

proper orthogonal decomposition closure models for turbulent flows a numerical comparison this paper puts forth two new closure models for the proper orthogonal decomposition reduced order modeling of structurally dominated turbulent flows the dynamic subgrid scale model and the variational multiscale model these models which are considered state of the art in large eddy simulation together with the mixing length and the smagorinsky closure models are tested in the numerical simulation of 3d turbulent flow past a circular cylinder at re 1000 five criteria are used to judge the performance of the proper orthogonal decomposition reduced order models the kinetic energy spectrum the mean velocity the reynolds stresses the root mean square values of the velocity fluctuations and the time evolution of the pod coefficients all the numerical results are benchmarked against a direct numerical simulation based on these numerical results we conclude that the dynamic subgrid scale and the variational multiscale models are the most accurate

centroidal voronoi tessellation based reduced order modeling of complex systems noindent a reduced order modeling methodology based on centroidal voronoi tessellations cvts is introduced cvts are special voronoi tessellations for which the generators of the voronoi diagram are also the centers of mass means of the corresponding voronoi cells for discrete data sets cvts are closely related to the h means and k means clustering techniques a discussion of reduced order modeling for complex systems such as fluid flows is given to provide a context for the application of reduced order bases then detailed descriptions of cvt based reduced order bases and how they can be constructed from snapshot sets and how they can be applied to the low cost simulation of complex systems are given subsequently some concrete incompressible flow examples are used to illustrate the construction and use of cvt based reduced order bases the cvt based reduced order modeling methodology is shown to be effective for these examples

a reduced order method for simulation and control of fluid flows this article presents a reduced order modeling approach for simulation and control of viscous incompressible flows the reduced order models suitable for control and which capture the essential physics are developed using the reduced basis method the so called lagrange approach is used to define reduced bases and the basis functions in this approach are obtained from the numerical solutions the feasibility of this method for flow control is demonstrated on boundary control problems in closed cavity and in wall bounded channel flows control action is effected through boundary surface movement on a part of the solid wall our formulation of the reduced order method applied to flow control problems leads to a constrained minimization problem and is solved by applying newton like methods to the necessary conditions of optimality through our computational experiments we demonstrate the feasibility and applicability of the reduced order method for simulation and control of fluid flows

pod galerkin reduced order methods for combined navier stokes transport equations based on a hybrid fv fe solver abstract the purpose of this work is to introduce a novel pod galerkin strategy for the semi implicit hybrid high order finite volume finite element solver introduced in bermudez et al 2014 and busto et al 2018 the interest is into the incompressible navier stokes equations coupled with an additional transport equation the full order model employed in this article makes use of staggered meshes this feature will be conveyed to the reduced order model leading to the definition of reduced basis spaces in both meshes the reduced order model presented herein accounts for velocity pressure and a transport related variable the pressure term at both the full order and the reduced order level is reconstructed making use of a projection method more precisely a poisson equation for pressure is considered within the reduced order model results are verified against three dimensional manufactured test cases moreover a modified version of the classical cavity test benchmark including the transport of a species is analysed

a bezoutian approach to orthogonal decompositions of trace forms or integral trace forms of some classical polynomials abstract as helmke and fuhrmann pointed out bezoutian approaches have been considered to be fruitful for the study of trace forms in this article we study orthogonal decompositions of trace forms or integral trace forms of some classical polynomials via bezoutians in section 3 we give another proof of a theorem of feit about orthogonal decompositions of trace forms of generalized laguerre polynomials in section 4 we consider integral trace forms of certain irreducible trinomials and give their orthogonal decompositions explicitly then in section 5 we obtain their canonical forms over z p the ring of p adic integers

on continuation of inviscid vortex patches abstract this investigation concerns solutions of the steady state euler equations in two dimensions featuring finite area regions with constant vorticity embedded in a potential flow using elementary methods of the functional analysis we derive precise conditions under which such solutions can be uniquely continued with respect to their parameters valid also in the presence of the kutta condition concerning a fixed separation point our approach is based on the implicit function theorem and perturbation equations derived using shape differentiation methods these theoretical results are illustrated with careful numerical computations carried out using the steklov poincare method which show the existence of a global manifold of solutions connecting the point vortex and the prandtl batchelor solution each of which satisfies the kutta condition

nonlinear model reduction via discrete empirical interpolation a dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition pod method for constructing reduced order models for time dependent and or parametrized nonlinear partial differential equations pdes in the presence of a general nonlinearity the standard pod galerkin technique reduces dimension in the sense that far fewer variables are present but the complexity of evaluating the nonlinear term remains that of the original problem the original empirical interpolation method eim is a modification of pod that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by pod we propose a discrete empirical interpolation method deim a variant that is suitable for reducing the dimension of systems of ordinary differential equations odes of a certain type as presented here it is applicable to odes arising from finite difference discretization of time dependent pdes and or parametrically dependent steady state problems however the approach extends to arbitrary systems of nonlinear odes with minor modification our contribution is a greatly simplified description of the eim in a finite dimensional setting that possesses an error bound on the quality of approximation an application of deim to a finite difference discretization of the one dimensional fitzhugh nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long time integration that fully captures nonlinear limit cycle behavior we also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results

canonical forms for hermitian matrix pairs under strict equivalence and congruence after a brief historical review and an account of the canonical forms attributed to jordan and kronecker a systematic development is made of the simultaneous reduction of pairs of quadratic forms over the complex numbers and over the reals these reductions are by strict equivalence and by congruence and essentially complete proofs are presented some closely related results which can be derived from the canonical forms are also included they concern simultaneous diagonalization a new criterion for the existence of positive definite linear combinations of a pair of hermitian matrices and the canonical structures of matrices which are self adjoint in an indefinite inner product

goal oriented model constrained optimization for reduction of large scale systems optimization oriented reduced order models should target a particular output functional span an applicable range of dynamic and parametric inputs and respect the underlying governing equations of the system to achieve this goal we present an approach for determining a projection basis that uses a goal oriented model constrained optimization framework the mathematical framework permits consideration of general dynamical systems with general parametric variations and is applicable to both linear and nonlinear systems results for a simple linear model problem of the two dimensional heat equation demonstrate the ability of the goal oriented approach to target a particular output functional of interest application of the methodology to a more challenging example of a subsonic blade row governed by the unsteady euler flow equations shows a significant advantage of the new method over the proper orthogonal decomposition

conification construction for kahler manifolds and its application in c projective geometry abstract two kahler metrics on a complex manifold are called c projectively equivalent if their j planar curves coincide such curves are defined by the property that the acceleration is complex proportional to the velocity the degree of mobility of a kahler metric is the dimension of the space of metrics that are c projectively equivalent to it we give the list of all possible values of the degree of mobility of a simply connected kahler manifold by reducing the problem to the study of parallel hermitian 0 2 tensors on the conification of the manifold we also describe all such values for a kahler einstein metric we apply these results to describe all possible dimensions of the space of essential c projective vector fields of kahler and kahler einstein metrics we also show that two c projectively equivalent kahler einstein metrics of arbitrary signature on a closed manifold have constant holomorphic curvature or are affinely equivalent

enablers for robust pod models this paper focuses on improving the stability as well as the approximation properties of reduced order models rom based on proper orthogonal decomposition pod the rom is obtained by seeking a solution belonging to the pod subspace and that at the same time minimizes the navier stokes residuals we propose a modified rom that directly incorporates the pressure term in the model the rom is then stabilized making use of a method based on the fine scale equations an improvement of the pod solution subspace is performed thanks to an hybrid method that couples direct numerical simulations and reduced order model simulations the methods proposed are tested on the two dimensional confined square cylinder wake flow in laminar regime

calibrated reduced order pod galerkin system for fluid flow modelling to improve the behaviour of reduced order proper orthogonal decomposition pod galerkin systems two numerical methods are proposed these methods determine free parameters in the pod galerkin system from flow simulations via a minimization problem they give rise to linear systems and their computational costs are reasonable both methods are assessed for two flow configurations a two dimensional flow around a square cylinder for a reynolds number of 100 and a three dimensional flow past a backward facing step for a reynolds number of 7432 based on the step height and the streamwise velocity at the middle of the inlet for both configurations the methods are effective since accurate calibrated reduced order pod galerkin systems are obtained

equation free galerkin free pod assisted computation of incompressible flows we present a galerkin free proper orthogonal decomposition pod assisted computational methodology for numerical simulations of the long term dynamics of the incompressible navier stokes equations the approach is based on the equation free framework we use short appropriate initialized bursts of full direct numerical simulations dns of the navier stokes equations to observe estimate and accelerate through projective integration the evolution of the flow dynamics the main assumption is that the long term dynamics of the flow lie on a low dimensional attracting and invariant manifold which can be parametrized not necessarily spanned by a few pod basis functions we start with a discussion of the consistency and accuracy of the approach and then illustrate it through numerical examples two dimensional periodic and quasi periodic flows past a circular cylinder we demonstrate that the approach can successfully resolve complex flow dynamics at a reduced computational cost and that it can capture the long term asymptotic state of the flow in cases where traditional galerkin pod models fail the approach trades the overhead involved in developing pod galerkin and pod nonlinear galerkin codes for the repeated yet short and on demand use of an existing full dns simulator moreover since in this approach the pod modes are used to observe rather than span the true system dynamics the computation is much less sensitive than pod galerkin to values of the system parameters e g the reynolds number and the particular simulation data ensemble used to obtain the pod basis functions

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

variational multiscale residual based turbulence modeling for large eddy simulation of incompressible flows we present an les type variational multiscale theory of turbulence our approach derives completely from the incompressible navier stokes equations and does not employ any ad hoc devices such as eddy viscosities we tested the formulation on forced homogeneous isotropic turbulence and turbulent channel flows in the calculations we employed linear quadratic and cubic nurbs a dispersion analysis of simple model problems revealed nurbs elements to be superior to classical finite elements in approximating advective and diffusive processes which play a significant role in turbulence computations the numerical results are very good and confirm the viability of the theoretical framework

local normal forms for geodesically equivalent pseudo riemannian metrics this is an extended version of the submitted paper it contains all proofs with all details and also an appendix where we explain how can one construct a complex structure by a 1 1 tensor such that all its eigenvalues are not real and such that the nijenhuis torsion vanishes

supremizer stabilization of pod galerkin approximation of parametrized steady incompressible navier stokes equations in this work we present a stable proper orthogonal decomposition pod galerkin approximation for parametrized steady incompressible navier stokes equations with low reynolds number supremizers solutions are added to the reduced velocity space in order to obtain a stable reduced order system considering in particular the fulfillment of an inf sup condition the stability analysis is first carried out from a theoretical standpoint then confirmed by numerical tests performed on a parametrized two dimensional backward facing step flow

a versatile sharp interface immersed boundary method for incompressible flows with complex boundaries a sharp interface immersed boundary method for simulating incompressible viscous flow past three dimensional immersed bodies is described the method employs a multi dimensional ghost cell methodology to satisfy the boundary conditions on the immersed boundary and the method is designed to handle highly complex three dimensional stationary moving and or deforming bodies the complex immersed surfaces are represented by grids consisting of unstructured triangular elements while the flow is computed on non uniform cartesian grids the paper describes the salient features of the methodology with special emphasis on the immersed boundary treatment for stationary and moving boundaries simulations of a number of canonical two and three dimensional flows are used to verify the accuracy and fidelity of the solver over a range of reynolds numbers flow past suddenly accelerated bodies are used to validate the solver for moving boundary problems finally two cases inspired from biology with highly complex three dimensional bodies are simulated in order to demonstrate the versatility of the method

bernstein gelfand gelfand sequences the bernstein gelfand gelfand sequences extend the complexes of homogeneous vector bundles to curved cartan geometries

hyperbolic manifolds are geodesically rigid we show that if all geodesics of two non proportional metrics on a closed manifold coincide as unparameterized curves then the manifold has a finite fundamental group or admits a local product structure this implies that if the manifold admits a metric of negative sectional curvature then two metrics on the manifold have the same geodesics if and only if they are proportional

projection based reduced order models for a cut finite element method in parametrized domains abstract this work presents a reduced order modeling technique built on a high fidelity embedded mesh finite element method such methods and in particular the cutfem method are attractive in the generation of projection based reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains and in general to handle topological changes the combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems avoiding remeshing as well as the reference domain formulation often used in the reduced order modeling for boundary fitted finite element formulations the resulting novel methodology is presented on linear elliptic and stokes problems together with several test cases to assess its capability the role of a proper extension and transport of embedded solutions to a common background is analyzed in detail

conification of kahler and hyper kahler manifolds given a kahler manifold m endowed with a hamiltonian killing vector field z we construct a conical kahler manifold hat m such that m is recovered as a kahler quotient of hat m similarly given a hyper kahler manifold m g j1 j2 j3 endowed with a killing vector field z hamiltonian with respect to the kahler form of j1 and satisfying mathcal l zj 2 2j 3 we construct a hyper kahler cone hat m such that m is a certain hyper kahler quotient of hat m in this way we recover a theorem by haydys our work is motivated by the problem of relating the supergravity c map to the rigid c map we show that any hyper kahler manifold in the image of the c map admits a killing vector field with the above properties therefore it gives rise to a hyper kahler cone which in turn defines a quaternionic kahler manifold our results for the signature of the metric and the sign of the scalar curvature are consistent with what we know about the supergravity c map

proper orthogonal decomposition with supg stabilized isogeometric analysis for reduced order modelling of unsteady convection dominated convection diffusion reaction problems abstract we consider reduced order modelling of unsteady convection dominated convection diffusion reaction problems with proper orthogonal decomposition pod in combination with isogeometric analysis isogeometric analysis has potential advantages in exact geometry representations efficient mesh generation different h p and k refinements and smooth bspline nurbs basis functions in order to compensate the oscillations caused by the convection dominated effect the streamline upwind petrov galerkin supg stabilization method is used both in generation of snapshots and pod galerkin method based on the recent novel and promising discretization method isogeometric analysis we propose a new fully discrete supg stabilized scheme the associated numerical error features three components due to spatial discretization by isogeometric analysis with supg stabilization time discretization with the crank nicolson scheme and modes truncation by pod we show a priori error estimates of the fully discrete scheme and give suitable stabilization parameters numerically a variety of two and three dimensional benchmark tests and numerical examples are provided to show the effectiveness accuracy and efficiency of the reduced order modelling methods by virtue of potential advantages of isogeometric analysis

conformal killing forms on riemannian manifolds conformal killing forms are a natural generalization of conformal vector fields on riemannian manifolds they are defined as sections in the kernel of a conformally invariant first order differential operator we show the existence of conformal killing forms on nearly kahler and weak g2 manifolds moreover we give a complete description of special conformal killing forms a further result is a sharp upper bound on the dimension of the space of conformal killing forms

clifford structures on riemannian manifolds we introduce the notion of even clifford structures on riemannian manifolds which for rank r 2 and r 3 reduce to almost hermitian and quaternion hermitian structures respectively we give the complete classification of manifolds carrying parallel rank r even clifford structures kahler quaternion kahler and riemannian products of quaternion kahler manifolds for r 2 3 and 4 respectively several classes of 8 dimensional manifolds for 5 r 8 families of real complex and quaternionic grassmannians for r 8 6 and 5 respectively and rosenfeld s elliptic projective planes op 2 c o p 2 h o p 2 and o o p 2 which are symmetric spaces associated to the exceptional simple lie groups f4 e6 e7 and e8 for r 9 10 12 and 16 respectively as an application we classify all riemannian manifolds whose metric is bundle like along the curvature constancy distribution generalizing well known results in sasakian and 3 sasakian geometry 2000 mathematics subject classification primary 53c26 53c35 53c10 53c15