three dimensional solution adaptive grid generation on composite configurations the eagle 3d composite block grid code has been coupled with an implicit euler flow solver to generate solution adaptive grids for composite block configurations with an improved control function formulation the code has been tested with two complex composite configurations one of which an eight block finned body of revolution at transonic speeds is described here the solution adaptive grids obtained possess continuous slopes across block boundaries and an improved quality of aerodynamics simulation about complex geometries

tilting modules and tilting torsion theories abstract we generalize basic results about classical tilting modules and partial tilting modules to the infinite dimensional case over an arbitrary ring r the methods employed combine classical techniques of representation theory of finite dimensional algebras with new techniques of the theory of modules using a generalization of the bongartz lemma we characterize tilting torsion theories in mod r i e torsion theories induced by infinitely generated tilting modules we investigate lattices gen p p of torsion classes induced by partial tilting modules p applying our results to tilting torsion classes we prove a version of the brenner butler theorem and a generalization of the assem smalo theorem to the case when r is artinian

a noniterative method for the generation of orthogonal coordinates in doubly connected regions in this paper a noniterative method for the numerical generation of orthogonal curvilinear coordinates for plane annular regions between two arbitrary smooth closed curves has been developed the basic generating equation is the gaussian equation for a euclidean space under the constraint of orthogonality the resulting equation has been solved analytically for the case of isothermic coordinates a coordinate transformation then yields nonisothermic coordinates in which the outgoing coordinate can be distributed in any desired manner the method has been applied in many cases and these test results demonstrate that the proposed method can be readily applied in a variety of problems

boundary fitted coordinate systems for numerical solution of partial differential equations a review a comprehensive review of methods of numerically generating curvilinear coordinate systems with coordinate lines coincident with all boundary segments is given some general mathematical framework and error analysis common to such coordinate systems is also included the general categories of generating systems are those based on conformal mapping orthogonal systems nearly orthogonal systems systems produced as the solution of elliptic and hyperbolic partial differential equations and systems generated algebraically by interpolation among the boundaries also covered are the control of coordinate line spacing by functions embedded in the partial differential operators of the generating system and by subsequent stretching transformation dynamically adaptive coordinate systems coupled with the physical solution and time dependent systems that follow moving boundaries are treated references reporting experience using such coordinate systems are reviewed as well as those covering the system development

smashing localizations of rings of weak global dimension at most one abstract we show for a ring r of weak global dimension at most one that there is a bijection between the smashing subcategories of its derived category and the equivalence classes of homological epimorphisms starting in r if moreover r is commutative we prove that the compactly generated localizing subcategories correspond precisely to flat epimorphisms we also classify smashing localizations of the derived category of any valuation domain and provide an easy criterion for the telescope conjecture tc for any commutative ring of weak global dimension at most one as a consequence we show that the tc holds for any commutative von neumann regular ring r and it holds precisely for those prufer domains which are strongly discrete

applications of numerical conformal mapping abstract the application of conformal mapping methods to the solution of free surface flow problems is considered methods of numerical conformal mapping based on fourier series are extended to handle efficiently problems with time dependent boundaries they are shown to be practicable only for moderately distorted geometries extensions of the menikoff zemach method to breaking geometries are presented these latter methods are robust at quite large distortions but degrade prematurely in time dependent problems at amplitudes smaller than achieved by our recent vortex methods

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

two dimensional unsteady euler equation solver for arbitrarily shaped flow regions a new technique is described for solving supersonic fluid dynamics problems containing multiple regions of continuous flow each bounded by a permeable or impermeable surface region boundaries are in general arbitrarily shaped and time dependent discretization of such a region for solution by conventional finite difference procedures is accomplished using an elliptic solver which alleviates the dependence on a particular base coordinate system multiple regions are coupled together through the boundary conditions the technique has been applied to a variety of problems including a shock diffraction problem and supersonic flow over a pointed ogive

simple algebraic technique for nearly orthogonal grid generation rid generation is a very important aspect of solving computational fluid dynamics cfd problems where a set of partial differential equations has to be solved in a given domain it is desirable for the grid to have good orthogonality properties and the grid generator should be able to produce an arbitrary amount of clustering near the body surface when necessary generation of body fitted coordinates meeting these requirements using algebraic techniques are described by eiseman1 and smith 2 it is well known that the most efficient way of dividing the domain for external flow problems is by generating an o type grid for a twodimensional case or an o o topology grid for a three dimensional case since they give the maximum resolution near the body surface with the minimum number of grid points a simple algebraic technique has been developed for generating an o type grid or an o o topology grid for a geometry with sharp edges having a high grid density near the body surface while maintaining orthogonality the present method generates the grid by distributing spacings in the outward normal direction from a closed contour as a function of the rate of change of arc length in that direction this procedure when applied to subsequently generated contours evolves a nearly orthogonal grid with a circular outer boundary a similar procedure applied to a closed three dimensional surface produces a spherical outer boundary thereby generating an o o topology grid

two numerical methods for the conformal mapping of simply connected domains abstract two numerical methods are considered for the conformai mapping of a bounded simply connected domain onto the unit disc the two methods are respectively the bergman kernel method which has been described in 17 and the so called ritz method in this paper we indicate the close theoretical relationship of the two methods compare their computational efficiencies and present a number of practical applications of the approximate conformal maps

conservation form of the navier stokes equations in general nonsteady coordinates introduction r interest in the generation of general bodyoriented curvilinear coordinate systems for the purpose of solving the complete navier stokes system of equations for subsonic and supersonic flows has given rise to many forms of presentations of the equations both in conservative and nonconservative formulations based on the available solutions of the gasdynamic equations e g ref 5 the conservation law form of the equations seems definitely preferable particularly when shocks are present although the above statement cannot be repeated in a definitive sense for the case of viscous subsonic and supersonic flow prediction through the navier stokes equations nevertheless it is expected that the conservation law form may eventually be more acceptable for numerical purposes the purpose of this paper is to derive the conservation law form of the navier stokes equations in general nonsteady coordinate systems in a simple and direct fashion previous work on this subject has been done by mcvittie viviand and vinokur it will be shown in this note that the equations in the conservation law form can be obtained simply by a little manipulation of some standard vector and tensor formulas

application of adaptive grids to fluid flow problems with asymptotic solutions proper grid point location is important in the asymptotic finite difference solution of any fluid flow or heat transfer problem this paper uses a new technique that provides a simple way of moving the mesh points in physical space in order to reduce the error in the computed asymptotic solution relative to that obtained using a fixed mesh applications to fluid flow problems are presented including boundary layer flow and inviscid supersonic flow over cylinders and wedges with associated detached shocks the treatment of curved bound aries stationary and nonstationary boundaries and systems of partial differential equations is discussed significant error reductions are demonstrated he accuracy of the finite difference solution of a partial differential equation is dependent upon the suitability of the grid used in computing the solution it is clear that two solutions to the same problem will not have equal accuracy for the same number of mesh points if the points are located differently in each case the typical grid point allocation problem in computational fluid dynamics may be stated in the following way if a solution to a set of partial differential equations is required on a physical domain subject to prescribed boundary conditions how should a fixed number of grid points be distributed in order to assure that the solution obtained is the best the term best as used here implies that some measure of error has been used in the point allocation since the size and complexity of the typical problem attempted using computational methods have in creased dramatically it is imperative that grid points be used as efficiently as possible to minimize storage requirements the construction of finite difference grid systems has received a great deal of attention over the past five years several techniques have been proposed for producing ac ceptable body fitted mesh systems probably the most well known of these methods is that of thompson et al 1 2 in which a solution of laplace s equation produces the grid in physical space this technique is ideally suited to elliptic problems where a grid point allocation can be completed at the beginning of the calculation hindman et al 3 produced an adaptive grid scheme for time dependent flows the grid speeds in physical space were obtained by using the idea proposed by thompson thames and mastin but in ad dition differentiating the equations for physical grid point location with respect to time at each time step a matrix inversion led to expressions for the components of the grid point velocity unfortunately hindman s work included only the effect of boundary motion on the grid speeds the technique used to determine grid point motion in the present study is that proposed by rai and anderson 4 the influence of both boundary motion and the details of the internal flowfield are included in the calculation of the grid point distribution numerical solutions for a number of fluid flow problems are presented showing the applicability of the technique to both viscous and inviscid flows these examples

silting and cosilting classes in derived categories abstract an important result in tilting theory states that a class of modules over a ring is a tilting class if and only if it is the ext orthogonal class to a set of compact modules of bounded projective dimension moreover cotilting classes are precisely the resolving and definable subcategories of the module category whose ext orthogonal class has bounded injective dimension in this article we prove a derived counterpart of the statements above in the context of silting theory silting and cosilting complexes in the derived category of a ring generalise tilting and cotilting modules they give rise to subcategories of the derived category called silting and cosilting classes which are part of both a t structure and a co t structure we characterise these subcategories silting classes are precisely those which are intermediate and ext orthogonal classes to a set of compact objects and cosilting classes are precisely the cosuspended definable and co intermediate subcategories of the derived category

partial silting objects and smashing subcategories we study smashing subcategories of a triangulated category with coproducts via silting theory our main result states that for derived categories of dg modules over a non positive differential graded ring every compactly generated localising subcategory is generated by a partial silting object in particular every such smashing subcategory admits a silting t structure

complete modules and torsion modules suppose that r is a ring and that a is a chain complex over r inside the derived category of differential graded a modules there are naturally defined subcategories of a torsion objects and of a complete objects under a finiteness condition on a we develop a morita theory for these subcat egories find conceptual interpretations for some associated algebraic functors and in appropriate commutative situations identify the associated functors as local homology or local cohomology some of the results are suprising even in the case r z and a z p 1 introduction let r be a ring and r mod the derived category of chain complexes of left modules see section 1 2 we choose a fixed complex a which is perfect in other words isomorphic in mod to a complex of finite length in which the entries are finitely generated projective modules we de clare another complex n to be a trivial if hom a a0 0 where hom denotes the chain complex of morphisms in r mod going further we say that x is a torsion if hom x at 0 for all a trivial n9 and that x is a complete if homr n9x 0 for all a trivial n we then study the category ators of a torsion complexes and the category acomp of a complete complexes both of these are triangulated full subcategories of r mod it turns out that these categories are equivalent to one another and also equivalent to the derived category of dif ferential graded modules over the endomorphism complex of a we construct approximation functors celu r mod at0rs and a r mod acomp for an object m of r mod9 the complex celu m is a kind of a cellular approxi mation to m in the sense that it is the best approximation to m which can be cobbled together from a and its suspensions the complex ma is the bousfield localization of m with respect to a homology theory on r mod derived from a we provide algebraic formulas for the functors and find that the functors are related in interesting ways one of which involves an arithmetic square we also show that if cciia r has certain finite dimensionality properties then an object m of r mod is a torsion or a complete if and only if the homology groups 77 m individually satisfy appropriate torsion or completeness conditions suppose now that r is a commutative ring that 7 c r is a finitely generated ideal that a r i9 and that k is the associated koszul complex the complex a

an implicit boundary approach for viscous compressible high reynolds flows using a hybrid remeshed particle hydrodynamics method abstract we present an implicit boundary particle method with background mesh adaptation we use a brinkman penalisation to represent the boundary of the domain and a remeshed particle method to simulate viscous flow with high reynolds numbers a penalty term is added to the navier stokes equations to impose the boundary conditions the boundary conditions are enforced to a specific precision with no need to modify the numerical method or change the grid achieving an implicit approach for flow around complex boundaries geometries simulation the main idea of the brinkman penalisation method is to model the solid obstacle as a porous medium the governing equations for the compressible fluid and penalised navier stokes for the porous medium are solved simultaneously without need for interface conditions the accuracy of the method is tested for a number of benchmark problems starting with simple cases such as a periodic laminar flow in a channel poiseuille flow to more complex problems such as the lid driven cavity with high reynolds numbers the reflection and propagation of a compressible shock wave and finally swirling flow in a model two stroke diesel engine the remeshed particle mesh method coupled with brinkman penalisation provides a good quality simulation and the results are in agreement with analytical or reference solutions

numerical treatment of unsteady subsonic flow past an oscillating cascade a numerical procedure for determining unsteady subsonic flow past a finite deflection oscillating cascade is described based on the assumption of small amplitu de blade motions the unsteady flow is governed by linear equations with variable coefficients which depend on the underlying steady flow these equations are solved on a nonorthogonal body fitted and periodic grid which facilitates the implementation of blade wake and cascade periodicity conditions but disallows the use of standard difference approximations instead difference approximations are based on an implicit least squares development applicable on arbitrary grids this permits flexibility in the choice of difference neighbors and the simultaneous approximations of differential equation and boundary conditions at boundary points strategies which can significantly improve numerical accuracy sample results illustrating the effects of blade thickness motion frequency and inlet mach number on the unsteady response of staggered cascades are presented n unsteady aerodynamic analysis which accounts for the effects of mean flow deflection due to blade geometry and flow turning and is capable of providing efficient predictions of unsteady response coefficients is an important requirement for turbomachinery applications a general aerodynamic model which fully includes the effects of nonuniform mean or steady flow on a small disturbance unsteady flow has been formulated and described in refs 1 and 2 the steady flow is governed by the full potential equations and the unsteady flow is governed by linear equations with variable coefficients which depend on the underlying steady flow in contrast to classical linear theory an unsteady aerodynamic formulation which includes the effects of nonuniform mean flow requires numerical solutions of the governing steady and unsteady boundary value problems numerical procedures for determining steady subsonic3 or transonic 4 potential flow through cascades are currently available a solution procedure for the linear variable coefficient unsteady problem is described in the present paper since the unsteady aerodynamic model treats realistic cascade configurations the numerical approximation must take into account nonrectangular geometries generally approximations to differential equations in nonrectangula r regions are developed in one of two ways a mapping may be defined so that in mapped variables calculation points can be placed in a rectangular grid the transformed equations are then approximated using rectangular difference expressions in the usual way alternatively more general difference ap proximations may be defined so that differential equations can be approximated on nonrectangula r meshes in physical variables thompson thames and mastin5 have defined global mappings for arbitrary two dimensional bodies but these mappings are inappropriate for cascades since the resulting meshes generally do not allow for cascade periodicity ives and liutermoza4 have determined global mappings which transform infinite two dimensional cascade regions into rectangular computational regions while allowing for cascade periodicity however the necessary mappings are not presently known for all cascades particularly those with

an initial value approach to the production of discrete orthogonal coordinates abstract a numerical procedure is described for the orthogonalization of a discrete non orthogonal coordinate system in two dimensions one set of the non orthogonal coordinate lines is preserved by the orthogonalization procedure the other set is shifted by interpolation to form the new orthogonal system the procedure is based upon an initial value formulation with the non orthogonal coordinates as the independent variables a first order linear partial differential equation is derived which describes the orthogonal trajectories a discrete representation of the equation is suggested and an interpolation procedure is described examples of non orthogonal with orthogonalized meshes are given including closed and open coordinate lines the method may be used to generate boundary fitted coordinates for boundary value problems or eulerian time dependent codes it may also be used at each timestep in a waterbag or semi lagrangian code

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

conformal mappings for internal viscous flow problems abstract isolated occlusions in a straight channel or pipe can be described analytically in a fashion that allows efficient accurate computation of nonlinear flow fields through conformal transformations

retinal vessel segmentation using a probabilistic tracking method vessel structures such as retinal vasculature are important features for computer aided diagnosis in this paper a probabilistic tracking method is proposed to detect blood vessels in retinal images during the tracking process vessel edge points are detected iteratively using local grey level statistics and vessel s continuity properties at a given step a statistic sampling scheme is adopted to select a number of vessel edge points candidates in a local studying area local vessel s sectional intensity profiles are estimated by a gaussian shaped curve a bayesian method with the maximum a posteriori map probability criterion is then used to identify local vessel s structure and find out the edge points from these candidates evaluation is performed on both simulated vascular and real retinal images different geometric shapes and noise levels are used for computer simulated images whereas real retinal images from the review database are tested evaluation performance is done using the segmentation matching factor smf as a quality parameter our approach performed better when comparing it with sun s and chaudhuri s methods roc curves are also plotted showing effective detection of retinal blood vessels true positive rate with less false detection false positive rate than sun s method

the use of splines and singular functions in an integral equation method for conformal mapping we consider the integral equation method of symm for the conformal mapping of simply connected domains for the numerical solution we examine the use of spline functions of various degrees for the approximation of the source density in particular we consider ways for overcoming the difficulties associated with corner singularities for this we modify the spline approximation and in the neighborhood of each corner where a boundary singularity occurs we approximate by a function which reflects the main singular behaviour of the source density the singular functions are then blended with the splines which approximate on the remainder of the boundary so that the global approximating function has continuity of appropriate order at the transition points between the two types of approximation we show by means of numerical examples that such approximations overcome the difficulties associated with corner singularities and lead to numerical results of high accuracy

silting modules and ring epimorphisms abstract there are well known constructions relating ring epimorphisms and tilting modules the new notion of silting module provides a wider framework for studying this interplay to every partial silting module we associate a ring epimorphism which we describe explicitly as an idempotent quotient of the endomorphism ring of the bongartz completion for hereditary rings this assignment is used to parametrise homological ring epimorphisms by silting modules we further show that homological ring epimorphisms of a hereditary ring form a lattice which completes the poset of noncrossing partitions in the case of finite dimensional algebras

a mapped factored implicit scheme for the computation of duct and far field acoustics system that is boundary fitted with grid points that may be concentrated as desired on the boundary for high resolution use of such a general mapping procedure together with an explicit acoustic algorithm is described by white 3 for the analysis of noise propagation in a two dimensional variablearea duct a second objective is that the algorithm should not require excessive computer memory but at the same time should provide an accurate approximation of the acoustic field in order to satisfy these requirements a transient finite difference method together with a factored implicit procedure was chosen finally the procedure should allow the noise calculation to proceed for some distance beyond the duct exit so that an appropriate acoustic boundary condition may be specified this approach should be more accurate than terminating the calculation at the duct exit where only an approximate boundary condition can be specified also information on the noise levels and radiation patterns emanating from the duct exit will be provided in the sections that follow the algorithm is briefly described and the numerical results of a test case are presented

on one dimensional stretching functions for finite difference calculations abstract the class of one dimensional stretching functions used in finite difference calculations is studied for solutions containing a highly localized region of rapid variation simple criteria for a stretching function are derived using a truncation error analysis these criteria are used to investigate two types of stretching functions one is an interior stretching function for which the location and slope of an interior clustering region are specified the simplest such function satisfying the criteria is found to be one based on the inverse hyperbolic sine it was first employed by thomas et al aiaa j 10 1972 887 the other type of function is a two sided stretching function for which the arbitrary slopes at the two ends of the one dimensional interval are specified the simplest such general function is found to be one based on the inverse tangent the special case where the slopes were both equal and greater than one was first employed by roberts the general two sided function has many applications in the construction of finite difference grids examples of such applications are found in the listed references

universal localizations via silting we show that silting modules are closely related with localizations of rings more precisely every partial silting module gives rise to a localization at a set of maps between countably generated projective modules and conversely every universal localization in the sense of cohn and schofield arises in this way to establish these results we further explore the finite type classification of tilting classes and we use the morphism category to translate silting modules into tilting objects in particular we prove that silting modules are of finite type

the telescope conjecture for hereditary rings via ext orthogonal pairs abstract for the module category of a hereditary ring the ext orthogonal pairs of subcategories are studied for each ext orthogonal pair that is generated by a single module a 5 term exact sequence is constructed the pairs of finite type are characterized and two consequences for the class of hereditary rings are established homological epimorphisms and universal localizations coincide and the telescope conjecture for the derived category holds true however we present examples showing that neither of these two statements is true in general for rings of global dimension 2