moduli spaces of lumps on real projective space harmonic maps that minimise the dirichlet energy in their homotopy classes are known as lumps lump solutions on real projective space are explicitly given by rational maps subject to a certain symmetry requirement this has consequences for the behaviour of lumps and their symmetries an interesting feature is that the moduli space of charge three lumps is a 7 dimensional manifold of cohomogeneity one which can be described as a one parameter family of symmetry orbits of d2 symmetric maps in this paper we discuss the charge three moduli spaces of lumps from two perspectives discrete symmetries of lumps and the riemann hurwitz formula we then calculate the metric and find explicit formulas for various geometric quantities we also discuss the implications for lump decay

the geometry of r adaptive meshes generated using optimal transport methods the principles of mesh equidistribution and alignment play a fundamental role in the design of adaptive methods and a metric tensor and mesh metric are useful theoretical tools for understanding a method s level of mesh alignment or anisotropy we consider a mesh redistribution method based on the monge ampere equation which combines equidistribution of a given scalar density function with optimal transport it does not involve explicit use of a metric tensor although such a tensor must exist for the method and an interesting question to ask is whether or not the alignment produced by the metric gives an anisotropic mesh for model problems with a linear feature and with a radially symmetric feature we derive the exact form of the metric which involves expressions for its eigenvalues and eigenvectors the eigenvectors are shown to be orthogonal and tangential to the feature and the ratio of the eigenvalues corresponding to the level of anisotropy is shown to depend both locally and globally on the value of the density function and the amount of curvature we thereby demonstrate how the optimal transport method produces an anisotropic mesh along a given feature while equidistributing a suitably chosen scalar density function numerical results are given to verify these results and to demonstrate how the analysis is useful for problems involving more complex features including for a non trivial time dependant nonlinear pde which evolves narrow and curved reaction fronts

a moving mesh method for the solution of the one dimensional phase field equations a moving mesh method is developed for the numerical solution of one dimensional phase change problems modelled by the phase field equations the computational mesh is obtained by equidistribution of a monitor function tailored for the functional variation of the phase field in the interfacial region existence and uniqueness of the discretised equations using a moving mesh are also established numerical results are given for classical and modified stefan test problems the numerical algorithm is relatively simple and is shown to be far more efficient than fixed grid methods

numerical solution of fisher s equation using a moving mesh method the paper investigates the viability of using moving mesh methods to simulate travelling wave solutions of fisher s equation results are presented that illustrate the weaknesses in moving mesh methods based on equidistribution of some popular monitor functions it is shown that knowledge of the differential equation and the travelling wave solution may be used to construct a monitor function that yields accurate results with suitably chosen moving mesh methods a comparison is made between a moving mesh partial differential equation and a moving mesh differential algebraic equation for the evolution in time

a moving finite element method with error estimation and refinement for one dimensional time dependent partial differential equations we discuss a moving finite element method for solving vector systems of time dependent partial differential equations in one space dimension the mesh is moved so as to equidistribute the spatial component of the discretization error in h 1 we present a method of estimating this error by using p hierarchic finite elements the error estimate is also used in an adaptive mesh refinement procedure to give an algorithm that combines mesh movement and refinement we discretize the partial differential equations in space using a galerkin procedure with piecewise linear elements to approximate the solution and quadratic elements to estimate the error a system of ordinary differential equations for mesh velocities are used to control element motions we use existing software for stiff ordinary differential equations for the temporal integration of the solution the error estimate and the mesh motion computational results using a code based on our method are presented for several examples

exact moduli space metrics for hyperbolic vortex polygons exact metrics on some totally geodesic submanifolds of the moduli space of static hyperbolic n vortices are derived these submanifolds denoted as n m are spaces of cn invariant vortex configurations with n single vortices at the vertices of a regular polygon and m n n coincident vortices at the polygon s center the geometric properties of n m are investigated and it is found that n n 1 is isometric to the hyperbolic plane of curvature 3 n 1 the geodesic flow on n m and a geometrically natural variant of geodesic flow recently proposed by collie and tong the dynamics of chern simons vortices phys rev d part fields gravit cosmol 78 065013 2008 e print arxiv hep th 0805 0602 are analyzed in detail

moving mesh methods for problems with blow up in this paper we consider the numerical solution of pdes with blow up for which scaling invariance plays a natural role in describing the underlying solution structures it is a challenging numerical problem to capture the qualitative behaviour in the blow up region and the use of nonuniform meshes is essential we consider moving mesh methods for which the mesh is determined using so called moving mesh partial differential equations mmpdes specifically the underlying pde and the mmpde are solved for the blow up solution and the computational mesh simultaneously motivated by the desire for the mmpde to preserve the scaling invariance of the underlying problem we study the effect of different choices of mmpdes and monitor functions it is shown that for suitable ones the mmpde solution evolves towards a moving mesh which close to the blow up point automatically places the mesh points in such a manner that the ignition kernel which is well known to be a natural coordinate in describing the behaviour of blow up approaches a constant as t to t the blow up time several numerical examples are given to verify the theory for these mmpde methods and to illustrate their efficacy

moving mesh partial differential equations to describe nematic order dynamics nematic liquid crystals are aggregates of calamitic molecules and most related experimental phenomena are described well by their mean molecular orientation i e by the director and by the scalar order parameter considering a perfect uniaxial symmetry however when the nematic distortion is very strong and it occurs over a length scale comparable with the nematic coherence length the molecular order may be significantly altered as in the case of the core of a defect or in the case of highly frustrated nematic systems such systems where spatial and or temporal changes of the nematic order are relevant require a full landau de gennes q tensor description in this work we will present the implementation of a q tensor numerical model based on a one dimensional finite element method with a r type moving mesh technique capable of describing the nematic order dynamics inside a p cell submitted to a strong electric pulse the use of the moving grid technique ensures no waste of computational effort in the area of low spatial order variability in fact the technique concentrates the grid points in regions of large q maintaining constant the total number of nodes in the domain

the kahler potential of abelian higgs vortices we calculate the kahler potential for the samols metric on the moduli space of abelian higgs vortices on r2 in two different ways the first uses a scaling argument the second depends on a variant of the relationship between accessory parameters and the regularized action in liouville field theory the kahler potential on the moduli space of vortices on h2 is also derived and we are led to a geometrical reinterpretation of these vortices

polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

lump dynamics in the ℂ p 1 model on the torus the topology and geometry of the moduli space m 2 of degree 2 static solutions of the p 1 model on a torus spacetime t 2 are studied it is proved that m 2 is homeomorphic to the left coset space g g 0 where g is a certain eight dimensional noncompact lie group and g 0 is a discrete subgroup of order 4 low energy two lump dynamics is approximated by geodesic motion on m 2 with respect to a metric g defined by the restriction to m 2 of the kinetic energy functional of the model this lump dynamics decouples into a trivial centre of mass motion and nontrivial relative motion on a reduced moduli space it is proved that m 2 g is geodesically incomplete and has only finite diameter a low dimensional geodesic submanifold is identified and a full description of its geodesics obtained

optimal mass transport for higher dimensional adaptive grid generation in this work we describe an approach for higher dimensional adaptive grid generation based on solving the l 2 monge kantorovich problem mkp which is a special case of the classical optimal mass transportation problem two methods are developed for computing the coordinate transformation used to define the grid adaptation for the first method the transformation is determined by solving a parabolic monge ampere equation for a steady state solution for the second method the grid movement is determined from the velocity field obtained by solving a fluid dynamics formulation of the l 2 mkp several numerical experiments are presented to demonstrate the performance of the mkp methods and to compare them with some related adaptive grid methods the experimental results demonstrate that the mkp methods show promise as effective and reliable methods for higher dimensional adaptive grid generation

quadratic duals koszul dual functors and applications this paper studies quadratic and koszul duality for modules over positively graded categories typical examples are modules over a path algebra which is graded by the path length of a not necessarily finite quiver with relations we present a very general definition of quadratic and koszul duality functors backed up by explicit examples this generalizes the work of beilinson ginzburg and soergel 1996 in two substantial ways we work in the setup of graded categories i e we allow infinitely many idempotents and also de ne a koszul duality functor for not necessarily koszul categories as an illustration of the techniques we reprove the koszul duality ryom hansen 2004 of translation and zuckerman functors for the classical category o in a quite elementary and explicit way from this we deduce a conjecture of bernstein frenkel and khovanov 1999 as applications we propose a definition of a koszul dual category for integral blocks of harish chandra bimodules and for blocks outside the critical hyperplanes for the kac moody category o

tilting and cluster tilting for quotient singularities we shall show that the stable categories of graded cohen macaulay modules over quotient singularities have tilting objects in particular these categories are triangle equivalent to derived categories of finite dimensional algebras our method is based on higher dimensional auslander reiten theory which gives cluster tilting objects in the stable categories of ungraded cohen macaulay modules

on the numerical solution of one dimensional pdes using adaptive methods based on equidistribution numerical experiments are described that illustrate some important features of the performance of moving mesh methods for solving one dimensional partial differential equations pdes the particular method considered here is an adaptive finite difference method based on the equidistribution of a monitor function and it is one of the moving mesh methods proposed by w huang y ren and r d russell 1994 siam j numer anal 31 709 we show how the accuracy of the computations is strongly dependent on the choice of monitor function and we present a monitor function that yields an optimal rate of convergence motivated by efficiency considerations for problems in two or more space dimensions we demonstrate a robust and efficient algorithm in which the mesh equations are uncoupled from the physical pde the accuracy and efficiency of the various formulations of the algorithm are considered and a novel automatic time step control mechanism is integrated into the scheme

simplicial structures in higher auslander reiten theory abstract we develop a novel combinatorial perspective on the higher auslander algebras of type a a family of algebras arising in the context of iyama s higher auslander reiten theory this approach reveals interesting simplicial structures hidden within the representation theory of these algebras and establishes direct connections to eilenberg maclane spaces and higher dimensional versions of waldhausen s s construction in algebraic k theory as an application of our techniques we provide a generalisation of the higher reflection functors of iyama and oppermann to representations with values in stable categories the resulting combinatorial framework of slice mutation can be regarded as a higher dimensional variant of the abstract representation theory of type a quivers developed by groth and s ovicek our simplicial point of view then naturally leads to an interplay between slice mutation horn filling conditions and the higher segal conditions of dyckerhoff and kapranov in this context we provide a classification of higher segal objects with values in any abelian category or stable category

moving mesh generation using the parabolic monge ampere equation this article considers a new method for generating a moving mesh which is suitable for the numerical solution of partial differential equations pdes in several spatial dimensions the mesh is obtained by taking the gradient of a scalar mesh potential function which satisfies an appropriate nonlinear parabolic partial differential equation this method gives a new technique for performing r adaptivity based on ideas from optimal transportation combined with the equidistribution principle applied to a time varying scalar monitor function used successfully in moving mesh methods in one dimension detailed analysis of this new method is presented in which the convergence regularity and stability of the mesh is studied additionally this new method is shown to be straightforward to program and implement requiring the solution of only one simple scalar time dependent equation in arbitrary dimension with adaptivity along the boundaries handled automatically we discuss three preexisting methods in the context of this work examples are presented in which either the monitor function is prescribed in advance or it is given by the solution of a partial differential equation

moving mesh methods based on moving mesh partial differential equations several versions of a moving mesh method are developed based on a mesh spatial smoothing technique and on the moving mesh pdes derived in a previous paper these versions are quite simple and easy to program they are applied to three bench mark one dimensional problems which show different solution behaviour the numerical results clearly demonstrate that the present methods are capable of accurately tracking rapid spatial and temporal transitions

an efficient numerical method for the solution of the l_2 optimal mass transfer problem in this paper we present a new computationally efficient numerical scheme for the minimizing flow approach for the computation of the optimal l 2 mass transport mapping in contrast to the integration of a time dependent partial differential equation proposed in s angenent s haker and a tannenbaum siam j math anal 35 2003 pp 61 97 we employ in the present work a direct variational method the efficacy of the approach is demonstrated on both real and synthetic data

analysis of moving mesh partial differential equations with spatial smoothing two moving mesh partial differential equations mmpdes with spatial smoothing are derived based upon the equidistribution principle this smoothing technique is motivated by the robust moving mesh method of dorfi and drury j comput phys 69 1987 pp 175 195 it is shown that under weak conditions the basic property of no node crossing is preserved by the spatial smoothing and a local quasi uniformity property of the coordinate transformations determined by these mmpdes is proven it is also shown that discretizing the mmpdes using centered finite differences these basic properties are preserved

a numerical study of three moving grid methods for one dimensional partial differential equations which are based on the method of lines in recent years several sophisticated packages based on the method of lines mol have been developed for the automatic numerical integration of time dependent problems in partial differential equations pdes notably for problems in one space dimension these packages greatly benefit from the very successful developments of automatic stiff ordinary differential equation solvers however from the pde point of view they integrate only in a semiautomatic way in the sense that they automatically adjust the time step sizes but use just a fixed space grid chosen a priori for the entire calculation for solutions possessing sharp spatial transitions that move e g travelling wave fronts or emerging boundary and interior layers a grid held fixed for the entire calculation is computationally inefficient since for a good solution this grid often must contain a very large number of nodes in such cases methods which attempt automatically to adjust the sizes of both the space and the time steps are likely to be more successful in efficiently resolving critical regions of high spatial and temporal activity methods and codes that operate this way belong to the realm of adaptive or moving grid methods following the mol approach this paper is devoted to an evaluation and comparison mainly based on extensive numerical tests of three moving grid methods for 1d problems viz the finite element method of miller and co workers the method published by petzold and a method based on ideas adopted from dorii and drury our examination of these three methods is aimed at assessing which is the most suitable from the point of view of retaining the acknowledged features of reliability robustness and efficiency of the conventional mol approach therefore considerable attention is paid to the temporal performance of the methods

dispersionless limit of the noncommutative potential kp hierarchy and solutions of the pseudodual chiral model in 2 1 dimensions the usual dispersionless limit of the kp hierarchy does not work in the case where the dependent variable has values in a noncommutative e g matrix algebra passing over to the potential kp hierarchy there is a corresponding scaling limit in the noncommutative case which turns out to be the hierarchy of a pseudodual chiral model in 2 1 dimensions pseudodual to a hierarchy extending ward s modified integrable chiral model applying the scaling procedure to a method generating exact solutions of a matrix potential kp hierarchy from solutions of a matrix linear heat hierarchy leads to a corresponding method that generates exact solutions of the matrix dispersionless potential kp hierarchy i e the pseudodual chiral model hierarchy we use this result to construct classes of exact solutions of the su m pseudodual chiral model in 2 1 dimensions including various multiple lump configurations

equidistributing principles in moving finite element methods abstract recently miller and his co workers proposed a moving finite element method based on a least squares principle this was followed by a similar method by the present authors using a petrov galerkin approach in this paper the two methods are compared in particular it is shown that both methods move their nodes according to an approximate equidistributing principle this observation leads to a criterion for the placement of the nodes it is also shown that the penalty function designed by miller may also be used with the petrov galerkin method finally numerical examples are given illustrating the performance of the two methods

a finite volume scheme for the patlak keller segel chemotaxis model a finite volume method is presented to discretize the patlak keller segel pks modeling chemosensitive movements first we prove existence and uniqueness of a numerical solution to the proposed scheme then we give a priori estimates and establish a threshold on the initial mass for which we show that the numerical approximation converges to the solution to the pks system when the initial mass is lower than this threshold numerical simulations are performed to verify accuracy and the properties of the scheme finally in the last section we investigate blow up of the solution for large mass

simple adaptive grids for 1 d initial value problems abstract in solving 1 d initial value problems by implicit finite difference methods it is advantageous to use an adaptive grid to provide local regions of high resolution and allow larger time steps to be taken a simple method of constructing such grids is presented and the benefits are demonstrated by calculations of sod s shock tube problem g a sod j comput phys 27 1 1978 and of a supernova explosion

auslander gorenstein algebras and precluster tilting abstract we generalize the notions of n cluster tilting subcategories and selfinjective algebras into n precluster tilting subcategories and n selfinjective algebras where we show that a subcategory naturally associated to n precluster tilting subcategories has a higher auslander reiten theory furthermore we give a bijection between n precluster tilting subcategories and n minimal auslander gorenstein algebras which is a higher dimensional analog of auslander solberg correspondence 8 as well as a gorenstein analog of n auslander correspondence 22 the auslander reiten theory associated to an n precluster tilting subcategory is used to classify the n minimal auslander gorenstein algebras into four disjoint classes our method is based on relative homological algebra due to auslander solberg

practical aspects of formulation and solution of moving mesh partial differential equations moving mesh partial differential equations mmpdes are used in the mmpde moving mesh method to generate adaptive moving meshes for the numerical solution of time dependent problems how mmpdes are formulated and solved is crucial to the efficiency and robustness of the method in this paper several practical aspects of formulating and solving mmpdes are studied they include spatial balance scaling invariance effective control of mesh concentration bounds on time steps multiple sub steps and two level mesh movement numerical results are also given c 2001 academic press

periodicity of d cluster tilted algebras abstract it is well known that any maximal cohen macaulay module over a hypersurface has a periodic free resolution of period 2 auslander reiten 1996 4 and buchweitz 1998 8 have used this periodicity to explain the existence of periodic projective resolutions over certain finite dimensional algebras which arise as stable endomorphism rings of cohen macaulay modules these algebras are in fact periodic meaning that they have periodic projective resolutions as bimodules and thus periodic hochschild cohomology as well the goal of this article is to generalize this construction of periodic algebras to the context of iyama s higher ar theory let c be a maximal d 1 orthogonal subcategory of an exact frobenius category b and start by studying the projective resolutions of finitely presented functors on the stable category c over both c and c under the assumption that c is fixed by d we show that d induces the 2 d th syzygy on mod c if c has finite type i e if c add t for a d cluster tilting object t then we show that the stable endomorphism ring of t has a quasi periodic resolution over its enveloping algebra moreover this resolution will be periodic if some power of d is isomorphic to the identity on c it follows in particular that 2 c y tilted algebras arising as stable endomorphism rings of cohen macaulay modules over curve singularities as in the work of burban iyama keller and reiten 2008 9 have periodic bimodule resolutions of period 4

on the time continuous mass transport problem and its approximation by augmented lagrangian techniques in j d benamou and y brenier numer math 84 2000 pp 375 393 a computational fluid dynamic approach was introduced for computing the optimal map occurring in the monge kantorovich problem though the described augmented lagrangian method involves a hilbertian framework the discussion was purely formal taking advantage of the recent progress in optimal transport theory l a caffarelli comm pure appl math 45 1992 pp 1141 1151 l a caffarelli ann of math 2 144 1996 pp 453 496 d cordero erausquin c r acad sci paris ser i math 329 1999 pp 199 202 r j mccann geom funct anal 11 2001 pp 589 608 and despite the lack of coercivity of the hilbertian problem we establish an existence result then under a reasonable assumption of positivity for the density we prove the existence of saddle points for both lagrangians defined in benamou and brenier and finally prove the convergence of the numerical method

homological theory of recollements of abelian categories abstract we investigate several homological aspects of recollements of abelian categories in particular we study how various homological invariants and dimensions of the categories involved in a recollement situation are related and when recollements of abelian categories induce recollements at the level of the bounded derived categories finally we give applications to global finitistic and representation dimension of rings and artin algebras and to rouquier s dimension of triangulated categories

higher nakayama algebras i construction we introduce higher dimensional analogues of the nakayama algebras from the viewpoint of iyama s higher auslander reiten theory more precisely for each nakayama algebra a and each positive intege

adaptive grid modelling for cancer cells in the early stage of invasion modelling the interaction of the urokinase plasminogen activator system with a model for cancer cells dynamics lead to a system of five coupled partial differential equations the solution of which over a one dimensional domain can be obtained using the finite element method as the discretization of the integration domain is crucial particularly in the presence of strong variations of the domain variables we tackle the problem solution implementing an adaptive grid numerical technique our results improve previous numerical simulations performed on uniformly discretized domains allowing to capture finer spatial details characterizing the interface between cancer and healthy cells which can be related to malignancy low values of the cancer cells motility induce a high spatial heterogeneity at the cancer healthy cells interface while the dynamical evolution over the whole domain shows branched spatio temporal patterns on the contrary higher motility values smooth the cancer cells spatial profile letting cancer cell concentration to evolve according to a less heterogeneous pattern

moving mesh strategy based on a gradient flow equation for two dimensional problems in this paper we introduce a moving mesh method for solving pdes in two dimensions it can be viewed as a higher dimensional generalization of the moving mesh pde mmpde strategy developed in our previous work for one dimensional problems w huang y ren and r d russell siam j numer anal 31 1994 pp 709 730 the mmpde is derived from a gradient flow equation which arises using a mesh adaptation functional in turn motivated from the theory of harmonic maps geometrical interpretations are given for the gradient equation and functional and basic properties of this mmpde are discussed numerical examples are presented where the method is used both for mesh generation and for solving time dependent pdes the results demonstrate the potential of the mesh movement strategy to concentrate the mesh points so as to adapt to special problem features and to also preserve a suitable level of mesh orthogonality

a moving mesh study for diffusion induced effects in avascular tumour growth abstract studying the dynamical evolution of complex systems as biological ones from the continuum point of view requires monitoring several parameters involved whose modelling leads to system of non linear coupled partial differential equations the interaction of the urokinase plasminogen activator system with a model for cancer cell in the avascular phase is faced with the moving mesh partial differential equation numerical technique monitoring the dynamical evolution of the system as a function of the diffusion properties of cancer cells and of cell proliferation factor over a one dimensional biological domain the computations are consistent with previous results confirming that cancer proliferation in the very early stage of invasion occurs through highly irregular spatio temporal pattern which depends essentially on cancer motility characteristics but non obvious effects are observed which depend on the model proliferation parameters

precise computations of chemotactic collapse using moving mesh methods we consider the problem of computing blow up solutions of chemotaxis systems or the so called chemotactic collapse in two spatial dimensions such solutions can have approximate self similar behaviour which can be very challenging to verify in numerical simulations cf betterton and brenner collapsing bacterial cylinders phys rev e 64 2001 061904 we analyse a dynamic scale invariant remeshing method which performs spatial mesh movement based upon equidistribution using a suitably chosen monitor function the numerical solution resolves the fine detail in the asymptotic solution structure such that the computations are seen to be fully consistent with the asymptotic description of the collapse phenomenon given by herrero and velazquez singularity patterns in a chemotaxis model math ann 306 1996 583 623 we believe that the methods we construct are ideally suited to a large number of problems in mathematical biology for which collapse phenomena are expected

galerkin finite element method for cancer invasion mathematical model abstract a finite element scheme for the solution of a cancer invasion model is proposed the cancer dynamics model consists of three coupled partial differential equations which describe the evolution of cancer cell density extra cellular matrix and the matrix degrading enzymes the model incorporates proliferation and haptotaxis effect of cancer cells their interaction with extracellular matrix the production of matrix degrading enzymes and consequent degradation of the extracellular matrix the coupled partial differential equations are discretized in space with the standard galerkin finite elements and in time with the crank nicolson method moreover the nonlinear terms in the coupled equations are treated semi implicitly in the finite element scheme the numerical scheme is validated with numerical results taken from the literature in addition to the mesh convergence study the effects of haptotactic rate proliferation rate and remodelling rate of matrix components of the considered mathematical model are investigated

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

stability of classical solutions of two dimensional grassmannian models we show that the only finite action solutions of the two dimensional grassmannian model that are stable under small fluctuations are the anti instanton solutions

the geodesic approximation for lump dynamics and coercivity of the hessian for harmonic maps the most fruitful approach to studying low energy soliton dynamics in field theories of bogomol nyi type is the geodesic approximation of manton in the case of vortices and monopoles stuart has obtained rigorous estimates of the errors in this approximation and hence proved that it is valid in the low speed regime his method employs energy estimates which rely on a key coercivity property of the hessian of the energy functional of the theory under consideration in this article we prove an analogous coercivity property for the hessian of the energy functional of a general sigma model with compact kahler domain and target we go on to prove a continuity property for our result and show that for the cp1 model on s2 the hessian fails to be globally coercive in the degree 1 sector we present numerical evidence which suggests that the hessian is globally coercive in a certain equivariance class of the degree n sector for n 2 we also prove that within the geodesic approximation a single cp1 lump movi

meshfree finite difference approximations for functions of the eigenvalues of the hessian we introduce meshfree finite difference methods for approximating nonlinear elliptic operators that depend on second directional derivatives or the eigenvalues of the hessian approximations are defined on unstructured point clouds which allows for very complicated domains and a non uniform distribution of discretisation points the schemes are monotone which ensures that they converge to the viscosity solution of the underlying pde as long as the equation has a comparison principle numerical experiments demonstrate convergence for a variety of equations including problems posed on random point clouds complex domains degenerate equations and singular solutions

the monge ampere trajectory correction for semi lagrangian schemes requiring that numerical estimates of the flow trajectories comply with the fundamental euler expansion formula that governs the evolution of a volume of fluid leads to a second order nonlinear monge ampere partial differential equation mae in cossette and smolarkiewicz 2011 15 a numerical algorithm based on solving the mae with an inexact newton krylov solver has been developed and used to correct standard estimates of the departure points of flow trajectories provided by a classical semi lagrangian scheme in the context of an incompressible fluid here we extend the theoretical analysis of the elemental rotational and deformational motions presented in cossette and smolarkiewicz 2011 15 in particular closed form analytic solutions are derived for both cases that serve to illustrate the mechanics of the enhanced trajectory scheme and to address the issues of existence and uniqueness scalar advection shows that the ma correction improves mass conservation substantially by suppressing anomalous fluid contraction the impact of the ma correction on complex flows is studied in the framework of an ideal magneto fluid in which the formation of current sheets leads to topological changes and reconnection of field lines the use of the ma correction improves the numerical stability of the solutions and prevents trajectory intersections as well as the spurious growth of the magnetic energy 2d and 3d examples are presented and the computational performance of the solver is documented

n representation infinite algebras abstract from the viewpoint of higher dimensional auslander reiten theory we introduce a new class of finite dimensional algebras of global dimension n which we call n representation infinite they are a certain analog of representation infinite hereditary algebras and we study three important classes of modules n preprojective n preinjective and n regular modules we observe that their homological behaviour is quite interesting for instance they provide first examples of algebras having infinite ext 1 orthogonal families of modules moreover we give general constructions of n representation infinite algebras applying minamoto s theory on fano algebras in non commutative algebraic geometry we describe the category of n regular modules in terms of the corresponding preprojective algebra then we introduce n representation tame algebras and show that the category of n regular modules decomposes into the categories of finite dimensional modules over localizations of the preprojective algebra this generalizes the classical description of regular modules over tame hereditary algebras as an application we show that the representation dimension of an n representation tame algebra is at least n 2

low energy dynamics of a cp1 lump on the sphere low energy dynamics in the unit charge sector of the cp1 model on spherical space space time s2 r is treated in the approximation of geodesic motion on the moduli space of static solutions a six dimensional manifold with nontrivial topology and metric the structure of the induced metric is restricted by consideration of the isometry group inherited from global symmetries of the full field theory evaluation of the metric is then reduced to finding five functions of one coordinate which may be done explicitly some totally geodesic submanifolds are found and the qualitative features of motion on these described

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

measuring mesh qualities and application to variational mesh adaptation the mesh assessment problem is investigated in this paper by taking into account the shape and size of elements and the solution behavior three elementwise mesh quality measures characterizing the shape alignment and adaptation features of elements are introduced according to the estimates of interpolation error developed on a general mesh an adaptive mesh is assessed by an overall quality measure defined as a weighted lebesgue norm of a product of the three elementwise quality measures it is shown that the overall quality of a mesh is good if the overall mesh quality measure is small or significantly smaller than the so called roughness measure of the solution defined as the ratio of two lebesgue norms of a derivative of the solution the definition of the overall mesh quality measure comes in such a way that the measure appears in the underlying error bound as the only factor depending substantially on the mesh as an immediate result the task of mesh adaptation becomes to control the overall mesh quality this idea is applied to variational mesh adaptation to develop two functionals one new and the other related to an existing functional recently developed using the regularity and equidistribution arguments numerical experiments are given to demonstrate the ability of the functionals to generate adaptive meshes of good quality

perpendicular categories with applications to representations and sheaves this paper is concerned with the omnipresence of the formation of the subcategories right left perpendicular to a subcategory of objects in an abelian category we encounter these subcategories in various contexts the formation of quotient categories with respect to localizing subcategories cf section 2 the deletion of vertices and shrinking of arrows see 37 in the representation theory of finite dimensional algebras cf section 5 the comparison of the representation theories of different extended dynkin quivers cf section 10 the theory of tilting cf sections 4 and 6 the study of homological epimorphisms of rings cf section 4 the passage from graded modules to coherent sheaves on a possibly weighted projective variety or scheme cf section 7 and 21 the study of maximal cohen macaulay modules over surface singularities cf sections the comparison of weighted projective lines for different weight sequences cf section 9 the formation of affine and local algebras attached to path algebras of extended dynkin quivers canonical algebras and weighted projective lines cf section 11 and 21 and the concept of universal localization in 40 formation of the perpendicular category has many aspects in common with localization and allows one to dispose of localization techniques in situations not accessible to any of the classical concepts of localization this applies in particular to applications in the domain of finite dimensional algebras and their representations several applications of the methods presented in this paper are already in existence partly published or appearing in print in the near future see for instance 40 39 4 45 26 49 46 and have show the versatility of the notion of a perpendicular category it seems that right perpendicular categories first appeared as the subcategories of so called closed objects in the process of the formation of the quotient cateogory of an abelian category with respect to a localizing serre subcategory see 18 47 34 another natural occurrence is encountered in commutative algebra forming the possibly infinitely generated modules of depth 2 cf section 7 the concept and some of the central applications were first presented in a talk given by the first author at the honnef meeting in january 1985 we also note that the perfectly matching nomination perpendicular category was coined by a schofield who discovered independently the usefulness of this concept in dealing with hereditary algebras see 39 cf also section 7 the authors further acknowledge the support of the deutsche forschungsgemeinschaft spp darstellungstheorie von endlichen gruppen und endlichdimensionalen algebren throughout this paper rings are associative with unit and modules are unitary right modules mod r respectively mod r denotes the category of all respectively all finitely presented right r modules

a simple adaptive grid method in two dimensions this paper gives an interpretation of the concept of equidistribution in the context of adaptive grid generation for multidimensional problems it is shown that the equidistribution principle cannot be satisfied throughout the domain of the problem and based on this recognition a local equidistribution principle is developed a discrete formulation is described for grid generation in two space dimensions and a smoothing mechanism is presented for improving mesh quality the adaptive grid method that is constructed contains three grid quality parameters numerical examples illustrate adaptive grid generation using a prescribed monitor function and grid generation for numerical solution of partial differential equations results show that the method produces high quality grids and that it is fairly insensitive to the choice of parameters

numerical simulation of blowup in nonlocal reaction diffusion equations using a moving mesh method in this paper we implement the moving mesh pde method for simulating the blowup in reaction diffusion equations with temporal and spacial nonlinear nonlocal terms by a time dependent transformation the physical equation is written into a lagrangian form with respect to the computational variables the time dependent transformation function satisfies a parabolic partial differential equation usually called moving mesh pde mmpde the transformed physical equation and mmpde are solved alternately by central finite difference method combined with a backward time stepping scheme the integration time steps are chosen to be adaptive to the blowup solution by employing a simple and efficient approach the monitor function in mmpdes plays a key role in the performance of the moving mesh pde method the dominance of equidistribution is utilized to select the monitor functions and a formal analysis is performed to check the principle a variety of numerical examples show that the blowup profiles can be expressed correctly in the computational coordinates and the blowup rates are determined by the tests

optimal discretizations in adaptive finite element electromagnetics one of the primary objectives of adaptive finite element analysis research is to determine how to effectively discretize a problem in order to obtain a sufficiently accurate solution efficiently therefore the characterization of optimal finite element solution properties could have significant implications on the development of improved adaptive solver technologies ultimately the analysis of optimally discretized systems in order to learn about ideal solution characteristics can lead to the design of better feedback refinement criteria for guiding practical adaptive solvers towards optimal solutions efficiently and reliably a theoretical framework for the qualitative and numerical study of optimal finite element solutions to differential equations of macroscopic electromagnetics is presented in this study for one two and three dimensional systems the formulation is based on variational aspects of optimal discretizations for helmholtz systems that are closely related to the underlying stationarity principle used in computing finite element solutions to continuum problems in addition the theory is adequately general and appropriate for the study of a range of electromagnetics problems including static and time harmonic phenomena moreover finite element discretizations with arbitrary distributions of element sizes and degrees of approximating functions are assumed so that the implications of the theory for practical h p hp and r type finite element adaption in multidimensional analyses may be examined copyright 2001 john wiley sons ltd

a positivity preserving adaptive moving mesh method for cancer cell invasion models abstract in this paper we present an efficient adaptive moving mesh finite difference method for the modeling of early stage cancer cell invasion of tissue or extracellular matrix ecm the cancer cell invasion model is a nonlinear system of reaction diffusion taxis partial differential equations pdes describing the time evolution of the cancer cells density and concentrations of proteins of the ecm the solution of the model exhibits very rapid variations at the boundary of the healthy and cancer cells thus using a uniform grid method to solve the cancer cell invasion model requires a very large number of grid points in order to resolve the steep gradients of the solution precisely as a result the computation can become prohibitively expensive and inefficient in this effort we propose a positivity preserving scheme for the spatial discretization of the model on an adaptive mesh to accurately capture the sharp structures of the solution the adaptive mesh is generated by a coordinate transformation obtained as the solution of the optimal mass transfer problem several numerical experiments are presented to demonstrate the performance of the adaptive mesh method for solving the cancer cell invasion model the numerical results show that the proposed adaptive mesh method is effective in capturing and predicting the correct behavior of the cancer cells movement within the ecm

an efficient dynamically adaptive mesh for potentially singular solutions abstract we develop an efficient dynamically adaptive mesh generator for time dependent problems in two or more dimensions the mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic pdes for the mesh map when coupled to a physical problem the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior the overall mesh strategy is simple to implement avoids interpolation and can be easily incorporated into a broad range of applications the efficacy of the mesh is first demonstrated by two examples of blowing up solutions to the 2 d semilinear heat equation these examples show that the mesh can follow with high adaptivity a finite time singularity process the focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2 d boussinesq fluid a challenging problem the moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically the numerical results show the fast collapse to small scales and an exponential vorticity growth

a moving mesh method based on the geometric conservation law a new adaptive mesh movement strategy is presented which unlike many existing moving mesh methods targets the mesh velocities rather than the mesh coordinates the mesh velocities are determined in a least squares framework by using the geometric conservation law specifying a form for the jacobian determinant of the coordinate transformation defining the mesh and employing a curl condition by relating the jacobian to a monitor function one is able to directly control the mesh concentration the geometric conservation law an identity satisfied by any nonsingular coordinate transformation is an important tool which has been used for many years in the engineering community to develop cell volume preserving finite volume schemes it is used here to transform the algebraic expression specifying the jacobian into an equivalent differential relation which is the key formula for the new mesh movement strategy it is shown that the resulting method bears a close relation with the lagrangian method advantages of the new approach include the ease of controlling the cell volumes and therefore mesh adaption and a theoretical guarantee for existence and nonsingularity of the coordinate transformation it is shown that the method may suffer from the mesh skewness a consequence resulting from its close relation with the lagrangian method numerical results are presented to demonstrate various features of the new method

optimal mass transport based adaptive mesh method for phase field models of two phase fluid flows in this paper we present an efficient adaptive mesh method for solving phase field model of a mixture of two phase incompressible fluid flows the adaptive mesh is generated by a coordinate transformation that is determined as the solution of the classical problem of the optimal mass transportation also known as mongekantorovich problem mkp the numerical solution of the mkp is computed by taking the gradient of the steady state solution of a parabolic mongeampre equation pmae several numerical experiments are conducted to demonstrate the performance of the pmae method for solving phase field models the numerical results illustrate that the adaptive mesh computed with pmae method captures well the moving interface of the phase field function

on the stability of mesh equidistribution strategies for time dependent partial differential equations abstract we study the stability of several mesh equidistribution schemes for time dependent partial differential equations in one space dimension the schemes move a finite difference or finite element mesh so that a given quantity is uniform over the domain we consider mesh moving methods that are based on solving a system of ordinary differential equations for the mesh velocities and show that some of these methods are unstable with respect to an equidistributing mesh when the partial differential system is dissipative using linear perturbation techniques we are able to develop simple criteria for determining the stability of a particular method and show how to construct stable differential systems for the mesh velocities several examples illustrating stable and unstable mesh motions are presented

sigma model solitons and their moduli space metrics we discuss the supersymmetric model and its soliton solutions in 2 1 dimensions we classify supersymmetric maps and derive bogomolny bounds we also give the modified superalgebra and describe the metric on the parameter space of solitons

higher dimensional auslander reiten theory on maximal orthogonal subcategories abstract we introduce the concept of maximal n 1 orthogonal subcategories over artin algebras and orders and develop n 1 dimensional auslander reiten theory on them we give the n auslander reiten translation and the n auslander reiten duality then show the existence of n almost split sequences and n fundamental sequences we give some examples

τ2 stable tilting complexes over weighted projective lines abstract let x be a weighted projective line and coh x the associated category of coherent sheaves we classify the tilting complexes t in d b coh x such that 2 t t where is the auslander reiten translation in d b coh x as an application of this result we classify the 2 representation finite algebras which are derived equivalent to a canonical algebra this complements iyama oppermann s classification of the iterated tilted 2 representation finite algebras by passing to 3 preprojective algebras we obtain a classification of the selfinjective cluster tilted algebras of canonical type this complements ringel s classification of the selfinjective cluster tilted algebras

d representation finite self injective algebras abstract in this paper we initiate the study of higher dimensional auslander reiten theory of self injective algebras we give a systematic construction of weakly d representation finite self injective algebras as orbit algebras of the repetitive categories of algebras of finite global dimension satisfying a certain finiteness condition for the serre functor the condition holds in particular for all fractionally calabi yau algebras of global dimension at most d this generalizes riedtmann s classical construction of representation finite self injective algebras our method is based on an adaptation of gabriel s covering theory for k linear categories to the setting of higher dimensional auslander reiten theory applications include n fold trivial extensions and classical and higher preprojective algebras which are shown to be d representation finite in many cases we also get a complete classification of all d representation finite self injective nakayama algebras for arbitrary d

quantitative analysis of cancer risk assessment in a mammalian cell with the inclusion of mitochondria abstract polycyclic aromatic hydrocarbons pahs are lipophilic environmental pollutants which are spread all over the world and are the cause of a number of toxic effects including carcinogenesis mutagenesis disruption of immune and endocrine function and developmental defects high lipid content of membranes of cell work as an open door for passing and amassing of lipophilic compounds amassing of pahs in cell damages the mitochondria and dna of the cell and hence gives rise to diseases like cancer to understand the correct intracellular metabolism and mobility of pahs a mathematical model for mitochondria in the form of both ordinary differential equations odes and partial differential equations pdes needs to be developed we developed a method using odes to replace the reaction diffusion system in a large number of mitochondria by one big mitochondrian pde model is developed to study more complex reaction diffusion system the numerical results at different input parameters show the inability of ode model to adequately describe the intracellular dynamics of pahs whereas the pde model behaved as it should it shows the inability of ode model to adequately reveal the essence of metabolism hence to study precise intracellular breakdown of pahs a modelling in terms of pde is required

maximal τd rigid pairs abstract let t be a 2 calabi yau triangulated category t a cluster tilting object with endomorphism algebra consider the functor t t t mod it induces a bijection from the isomorphism classes of cluster tilting objects to the isomorphism classes of support tilting pairs this is due to adachi iyama and reiten the notion of d 2 angulated categories is a higher analogue of triangulated categories we show a higher analogue of the above result based on the notion of maximal d rigid pairs

on the classification of higher auslander algebras for nakayama algebras abstract we give new improved bounds for the dominant dimension of nakayama algebras and use those bounds to give a classification of nakayama algebras with n simple modules that are higher auslander algebras with global dimension at least n the classification is then used to extend the results on the inequality for the global dimension of nakayama algebras obtained in 18

maximal modifications and auslander reiten duality for non isolated singularities we first generalize classical auslander reiten duality for isolated singularities to cover singularities with a one dimensional singular locus we then define the notion of ct modules for non isolated singularities and we show that these are intimately related to noncommutative crepant resolutions nccrs when r has isolated singularities ct modules recover the classical notion of cluster tilting modules but in general the two concepts differ then wanting to generalize the notion of nccrs to cover partial resolutions of operatorname spec r in the main body of this paper we introduce a theory of modifying and maximal modifying modules under mild assumptions all the corresponding endomorphism algebras of the maximal modifying modules for three dimensional gorenstein rings are shown to be derived equivalent we then develop a theory of mutation for modifying modules which is similar but different to mutations arising in cluster tilting theory our mutation works in arbitrary dimension and in dimension three the behavior of our mutation strongly depends on whether a certain factor algebra is artinian

n angulated categories we dene n angulated categories by modifying the axioms of triangulated categories in a natural way we show that heller s parametrization of pre triangulations extends to pre n angulations we obtain a large class of examples of n angulated cate gories by considering n 2 cluster tilting subcategories of triangulated categories which are stable under the n 2 nd power of the suspension functor as an application we show how n angulated calabi yau categories yield triangulated calabi yau categories of higher calabi yau dimension finally we sketch a link to algebraic geometry and string theory

the adiabatic limit of wave map flow on a two torus the s2 valued wave map flow on a lorentzian domain r where is any flat two torus is studied the cauchy problem with initial data tangent to the moduli space of holomorphic maps s2 is onsidered in the limit of small initial velocity it is proved that wave maps in this limit converge in a precise sense to geodesics in the moduli space of holomorphic maps with respect to the l2 metric this establishes in a rigorous setting a long standing informal conjecture of ward

n representation finite algebras and n apr tilting we introduce the notion of n representation finiteness generalizing representationfinite hereditary algebras we establish the procedure of n apr tilting and show that it preserves n representation finiteness we give some combinatorial description of this procedure and use this to completely describe a class of n representation finite algebras called type a

parabolic monge ampere methods for blow up problems in several spatial dimensions this paper constructs and analyses an adaptive moving mesh scheme for the numerical simulation of singular pdes in one or more spatial dimensions the scheme is based on computing a legendre transformation from a regular to a spatially non uniform mesh via the solution of a relaxed form of the monge ampere equation the method is shown to preserve the inherent scaling properties of the pde and to identify natural computational coordinates numerical examples are presented in one and two dimensions which demonstrate the effectiveness of this approach

noetherian hereditary abelian categories satisfying serre duality norwegian univ sci technol dept math sci n 7491 trondheim norway limburgs univ ctr dept wni b 3590 diepenbeek belgium reiten i norwegian univ sci technol dept math sci n 7491 trondheim norway