convolution generated motion and generalized huygens principles for interface motion a physical interface can often be modeled as a surface that moves with a velocity determined by the local geometry accordingly there is great interest in algorithms that generate such geometric interface motion in this paper we unify and generalize two simple algorithms for constant and mean curvature based interface motion the classical huygens principle and diffusion generated motion we show that the resulting generalization can be viewed both geometrically as a type of huygens principle and algebraically as a convolution generated motion using the geometric algebraic duality from the unification we construct specific convolution generated motion algorithms for a common class of anisotropic curvature dependent motionlaws we validate these algorithms with numerical experiments and show that they can be implemented accurately and efficiently with adaptive resolution and fast fourier transform techniques

symmetry and duality on n gorenstein rings abstract we will study homological properties of noetherian rings as a bridge between the n gorenstein property and the dominant dimension we will introduce the dominant numbers of rings and prove their left right symmetry section 1 1 the l n condition section 2 2 on selfinjective resolutions provides a useful tool on our study moreover we will give two dualities on n gorenstein rings sections 1 2 1 3 where one of them is closely related to the auslander reiten theory section 1 3 1

on the ideal theory of graphs we study algebras defined by finite sets g m1 mq of monomials of a polynomial ring r there are two basic algebras i k g k m1 mq the k subalgebra of r spanned by the mi and ii the quotient ring r i g where i g m1 mq they come together in the construction of the rees algebra r i g of the ideal i g the emphasis is almost entirely on sets of squarefree monomials of degree two and their attached graphs the main results are assertions about the cohen macaulay behaviour of the koszul homology of i g and how normality or cohen macaulayness of one of the algebras can be read off the properties of the graph or in the other algebra

geometric computation of curvature driven plane curve evolutions we present a new numerical scheme for planar curve evolution with a normal velocity equal to f kappa where kappa is the curvature and f is a nondecreasing function such that f 0 0 and either x mapsto f x 3 is lipschitz with lipschitz constant less than or equal to 1 or f x x gamma for gamma geq 1 3 the scheme is completely geometrical and avoids some drawbacks of finite difference schemes in particular no special parameterization is needed and the scheme is monotone that is if a curve initially surrounds another one then this remains true during their evolution which guarantees numerical stability we prove consistency and convergence of this scheme in a weak sense finally we display some numerical experiments on synthetic and real data

equality of ordinary and symbolic powers of stanley reisner ideals abstract this paper studies properties of simplicial complexes with the equality i m i m for a given m 2 the main results are combinatorial characterizations of such complexes in the two dimensional case it turns out that there exists only a finite number of complexes with this property and that these complexes can be described completely as a consequence we are able to determine all complexes for which i m is cohen macaulay for some m 2 in particular there are complexes with i 2 i 2 or i 3 i 3 but i m i m for all m 4 and that if i m i m for some m 4 then i m i m for all m 1 similarly there are complexes for which i 2 is cohen macaulay but i m is not cohen macaulay for all m 3 and if i m is cohen macaulay for some m 3 then i is a complete intersection

manifold evolution with tangential redistribution of points in this paper we propose several techniques for tangential redistribution of points on evolving surfaces this is an important issue in numerical approximation of any lagrangian evolution model since the quality of the mesh has a significant impact on the result of the computation we explain the volume oriented and length oriented tangential redistribution methods in a general setting of an m dimensional manifold evolving in an n dimensional manifold then we apply the proposed techniques to several manifold evolution problems we explain the numerical approximation of the models and present experiments illustrating the performance of our redistribution techniques

on affine plane curve evolution abstract an affine invariant curve evolution process is presented in this work the evolution studied is the affine analogue of the euclidean curve shortening flow evolution equations for both affine and euclidean invariants are developed an affine version of the classical euclidean isoperimetric inequality is proved this inequality is used to show that in the case of affine evolution of convex plane curves the affine isoperimetric ratio is a non decreasing function of time convergence of this affine isoperimetric ratio to the ellipse s value 8 2 as well as convergence in the hausdorff metric of the evolving curve to an ellipse is also proved

the existence of maximal n orthogonal subcategories abstract for an n 1 auslander algebra with global dimension n we give some necessary conditions for admitting a maximal n 1 orthogonal subcategory in terms of the properties of simple modules with projective dimension n 1 or n for an almost hereditary algebra with global dimension 2 we prove that admits a maximal 1 orthogonal subcategory if and only if for any non projective indecomposable module m m is injective is equivalent to that the reduced grade of m is equal to 2 we give a connection between the gorenstein symmetric conjecture and the existence of maximal n orthogonal subcategories of t for a cotilting module t for a gorenstein algebra we prove that all non projective direct summands of a maximal n orthogonal module are n periodic in addition we study the relation between the complexity of modules and the existence of maximal n orthogonal subcategories for the tensor product of two finite dimensional algebras

on the cohen macaulayness of the conormal module of an ideal abstract in the present paper we investigate a question stemming from a long standing conjecture of vasconcelos given a generically complete intersection perfect ideal i in a regular local ring r when is it true that the cohen macaulayness of i i 2 or r i 2 implies that r i is gorenstein this property is known to hold for licci ideals and essentially squarefree monomial ideals we show that a positive answer actually holds for every monomial ideal we then give a positive answer for several special classes of ideals and provide application to algebroid curves with low multiplicity we also exhibit prime ideals in regular local rings and homogeneous level ideals in polynomial rings for which the answer is negative and use them to show the sharpness of our main result as they lie in the first class of ideals not covered by it the homogeneous examples have been found thanks to the help of j c migliore as a by product we exhibit several classes of cohen macaulay ideals whose square is not cohen macaulay our methods work both in the homogeneous and in the local settings

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

on the affine heat equation for non convex curves 1998 american mathematical society first published in journal of the american mathematical society vol 11 no 3 july 1998 published by the american mathematical society

differential and numerically invariant signature curves applied to object recognition we introduce a new paradigm the differential invariant signature curve or manifold for the invariant recognition of visual objects a general theorem of e cartan implies that two curves are related by a group transformation if and only if their signature curves are identical the important examples of the euclidean and equi affine groups are discussed in detail secondly we show how a new approach to the numerical approximation of differential invariants based on suitable combination of joint invariants of the underlying group action allows one to numerically compute differential invariant signatures in a fully group invariant manner applications to a variety of fundamental issues in vision including detection of symmetries visual tracking and reconstruction of occlusions are discussed

affine plane curve evolution a fully consistent scheme we present an accurate numerical scheme for the affine plane curve evolution and its morphological extension to grey level images this scheme is based on the iteration of a nonlocal fully affine invariant and numerically stable operator which can be exactly computed on polygons the properties of this operator ensure that a few iterations are sufficient to achieve a very good accuracy unlike classical finite difference schemes that generally require a lot of iterations convergence results are provided as well as theoretical examples and experiments

some qualitative properties for geometric flows and its euler implicit discretization abstract we study the geometric flow parabolic equation and its implicit discretization which yield a family of nonlinear elliptic problems we show that there are important differences in the study of those equations which concerns the propagation of level sets of data our study is based on the previous study of radially symmetric solutions of the corresponding equation curiously in radial coordinates both equations reduce to suitable singular hamilton jacobi first order equations after considering the case of monotone data we point out a new peculiar behavior for non monotone data with a profile of batman type g min g 1 g 2 g 1 r increasing g 2 r decreasing and g 1 r d g 2 r d for some r d 0 in the parabolic regime and when the velocity of the convexity part of the level sets is greater than the velocity of the concavity part we show that the level set u g r d develops a non empty interior set for any t 0 nothing similar occurs in the stationary regime we also present some numerical experiences

tilting modules of finite projective dimension tilting modules are well known and are useful in the representation theory of artinian algebras 2 4 6 11 the purpose of this paper is to define tilting modules of finite projective dimension and prove some fundamental results a unital left a module a t over a ring a is said to be a tilting module of projective dimension 0 is an integer if it satisfies the following three conditions 1 a t has a projective resolution

koszul homology and the structure of low codimension cohen macaulay ideals the relationship between the properties of the koszul homology modules of two ideals connected by linkage is studied if the ideal i is either i a cohen macaulay ideal of codimension 3 or ii a gorenstein ideal of codimension 4 the one dimensional koszul module carries considerable information on the struct ural nature of the linkage class of i in case i or on the conormal module of i in case ii emphasis is given to the verification of the properties by computation introduction the aim of this paper is to examine the structure of cohen macaulay ideals of low i e 3 and 4 codimension as it is expressed by the koszul homology modules more specifically given a perfect ideal i of a gorenstein nng r we are interested in how i its numerical data e g betti numbers and degrees of its generating syzygies in the case of a homogeneous ideal ii properties of the full linkage class of i and iii whether i is suitably generated e g generated by a d sequence are mirrored on the module and the algebra structures of the homology of the koszul complex build on a generating set of i the man result theorem 2 4 is the proof that for a perfect ideal i of codimension threes the condition that the 1 dimensional koszul homology module h1 i r be cohen macaulay is an invanant of the full linkage class of i the framework for this result is provided by several facts first the classical descnptions of cohen macaulay ideals of codimension two and of gorenstein ideals of heit ht three as belonging to the linkage classes of complete intersections a be ga w then a theorem of huneke hu1 asserting that the property hi i r is cohen macaulay for i ms is an invariant of the eren linkage class of i finally in case of ideals of polynomial rings data on the koszul homology has the advantage of being in principle known at the outset that is hypotheses on them can be venfied once the ideal has been specified by a set of generators in the last section we indicate how symbolic computation packages may be used to obtain this informa tion to contrast already in the case of codimension four gorenstein ideals the depths of the koszul homology modules of directly linked ideals may differ sharply furthermore in codimension three we show that this invanance does not extend to the next koszul module to further enhance the role of h1 i r as a carrier of

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

jacobian criteria for complete intersections the graded case letp be a positively graded polynomial ring over a fieldk of characteristic zero leti be a homogeneous ideal ofp and setr p i the paper investigates the homological properties of somer modules canonically associated withr among them the module r k of kahler differentials and the conormal modulei i 2 it is shown that a subexponential bound on the betti numbers of any of these modules implies thati is generated by ap regular sequence in particular the finiteness of the projective dimension of the conormal module impliesr is a complete intersection similarly the finiteness of the projective dimension of the differential module impliesr is a reduced complete intersection this provides strong converses to some well known properties of complete intersections and establishes special cases of conjectures of vasconcelos the proofs of these results make extensive use of differential graded homological algebra the crucial step is to show that any homomorphism of complexes from the minimal cotangent complexl r k of andre and quillen into the minimal free resolution of the irrelevant maximal ideal m ofr which extends the euler map r k is a split embedding of gradedr modules

motion of level sets by mean curvature i we continue our investigation of the level set technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature the principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by k brakke we prove that almost every level set of the solution to the mean curvature evolution pde is in fact aunit density varifold moving according to its mean curvature in particular a e level set is endowed with a kind of geometric structure the proof utilizes compensated compactness methods to pass to limits in various geometric expressions

higher auslander algebras admitting trivial maximal orthogonal subcategories abstract in this paper we prove that is an n 1 auslander artinian algebra with gl dim n 2 admitting a trivial maximal n 1 orthogonal subcategory of mod if and only if it is morita equivalent to a finite product of f and f f 0 0 0 0 f f 0 0 0 0 f 0 0 0 0 0 f f 0 0 0 0 f n 1 n 1 where f is a division algebra in addition we obtain a necessary condition for an auslander artinian algebra admitting a non trivial maximal 1 orthogonal subcategory

curvature and the evolution of fronts the evolution of a front propagating along its normal vector field with speedf dependent on curvaturek is considered the change in total variation of the propagating front is shown to depend only ondf dk only wherek changes sign analysis of the casef k 1 ek where e is a constant shows that curvature plays a role similar to that of viscosity in burgers equation for e 0 and non convex initial data the curvature blows up corners develop and an entropy condition can be formulated to provide an explicit construction for a weak solution beyond the singularity we then numerically show that the solution as e goes to zero converges to the constructed weak solution numerical methods based on finite difference schemes for marker particles along the front are shown to be unstable in regions where the curvature builds as a remedy we show that front tracking based on volume of fluid techniques can be used together with the entropy condition to provide transition from the classical to weak solution

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

evolution of plane curves driven by a nonlinear function of curvature and anisotropy in this paper we study evolution of plane curves satisfying a geometric equation v beta k nu where v is the normal velocity and k and nu are the curvature and tangential angle of a plane curve gamma we follow the direct approach and we analyze the so called intrinsic heat equation governing the motion of plane curves obeying such a geometric equation the intrinsic heat equation is modified to include an appropriate nontrivial tangential velocity functional alpha we show how the presence of a nontrivial tangential velocity can prevent numerical solutions from forming various instabilities from an analytical point of view we present some new results on short time existence of a regular family of evolving curves in the degenerate case when beta k nu gamma nu k m 0 m le 2 and the governing system of equations includes a nontrivial tangential velocity functional

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

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