inseparable local uniformization abstract it is known since the works of zariski in the early 40s that desingularization of varieties along valuations called local uniformization of valuations can be considered as the local part of the desingularization problem it is still an open problem if local uniformization exists in positive characteristic and dimension larger than three in this paper we prove that zariski local uniformization of algebraic varieties is always possible after a purely inseparable extension of the field of rational functions and therefore any valuation can be uniformized by a purely inseparable alteration

multiplier ideals and modules on toric varieties a formula computing the multiplier ideal of a monomial ideal on an arbitrary affine toric variety is given variants for the multiplier module and test ideals are also treated

three dimensional ego motion estimation from motion fields observed with multiple cameras abstract in this paper we present a robust method to estimate the three dimensional ego motion of an observer moving in a static environment this method combines the optical flow fields observed with multiple cameras to avoid the ambiguity of 3 d motion recovery due to small field of view and small depth variation in the field of view two residual functions are proposed to estimate the ego motion for different situations in the non degenerate case both the direction and the scale of the three dimensional rotation and translation can be obtained in the degenerate case rotation can still be obtained but translation can only be obtained up to a scale factor both the number of cameras and the camera placement affect the accuracy of the estimated ego motion we compare different camera configurations through simulation some results of real world experiments are also given to demonstrate the benefits of our method

the tight integral closure of a set of ideals abstract this paper celebrates the enormous contributions of david buchsbaum to commutative algebra homological algebra and representation theory with grateful appreciation for the inspiration he has provided for myself and so many others

length multiplicity and multiplier ideals let r m be an n dimensional regular local ring essentially of finite type over a field of characteristic zero given an m primary ideal a of r the relationship between the singularities of the scheme defined by a and those defined by the multiplier ideals j a c with c varying in q are quantified in this paper by showing that the samuel multiplicity of a satisfies e n n k n c n whenever j a c c m k 1 this formula generalizes an inequality on log canonical thresholds previously obtained by ein mustata and the author of this paper a refined inequality is also shown to hold for small dimensions and similar results valid for a generalization of test ideals in positive characteristics are presented

a characterization of rational singularities in terms of injectivity of frobenius maps the notions of f rational and f regular rings are defined via tight closure which is a closure operation for ideals in a commutative ring of positive characteristic the geometric significance of these notions has persisted and k e smith proved that f rational rings have rational singularities we now ask about the converse implication the answer to this question is yes and no for a fixed positive characteristic there is a rational singularity which is not f rational so the answer is no in this paper however we aim to show that the answer is yes in the following sense if a ring of characteristic zero has rational singularity then its modulo p reduction is f rational for almost all characteristic p this result leads us to the correspondence of f regular rings and log terminal singularities

f singularities under generic linkage abstract let r k x 1 x n be a polynomial ring over a prefect field of positive characteristic let i be an equi dimensional ideal in r and let j be a generic link of i in s r u i j c r we describe the parameter test submodule of s j in terms of the test ideal of the pair r i when a reduction of i is a complete intersection or almost complete intersection as an application we deduce a criterion for when s j has f rational singularities in these cases we also compare the f pure threshold of r i and s j

estimation of object motion parameters from noisy images an approach is presented for the estimation of object motion parameters based on a sequence of noisy images the problem considered is that of a rigid body undergoing unknown rotational and translational motion the measurement data consists of a sequence of noisy image coordinates of two or more object correspondence points by modeling the object dynamics as a function of time estimates of the model parameters including motion parameters can be extracted from the data using recursive and or batch techniques this permits a desired degree of smoothing to be achieved through the use of an arbitrarily large number of images some assumptions regarding object structure are presently made results are presented for a recursive estimation procedure the case considered here is that of a sequence of one dimensional images of a two dimensional object thus the object moves in one transverse dimension and in depth preserving the fundamental ambiguity of the central projection image model loss of depth information an iterated extended kalman filter is used for the recursive solution noise levels of 5 10 percent of the object image size are used approximate cramer rao lower bounds are derived for the model parameter estimates as a function of object trajectory and noise level this approach may be of use in situations where it is difficult to resolve large numbers of object match points but relatively long sequences of images 10 to 20 or more are available

structure from motion causally integrated over time we describe an algorithm for reconstructing three dimensional structure and motion causally in real time from monocular sequences of images we prove that the algorithm is minimal and stable in the sense that the estimation error remains bounded with probability one throughout a sequence of arbitrary length we discuss a scheme for handling occlusions point features appearing and disappearing and drift in the scale factor these issues are crucial for the algorithm to operate in real time on real scenes we describe in detail the implementation of the algorithm which runs on a personal computer and has been made available to the community we report the performance of our implementation on a few representative long sequences of real and synthetic images the algorithm which has been tested extensively over the course of the past few years exhibits honest performance when the scene contains at least 20 40 points with high contrast when the relative motion is slow compared to the sampling frequency of the frame grabber 30 hz and the lens aperture is large enough typically more than 30 spl deg of visual field

optimal visual motion estimation a note the problem of estimating 3d motion in an optimal manner using correspondences of features in two views is analyzed the importance of having an optimal estimator is twofold first for the estimation itself and second for the bound it offers on how much sensitivity one can expect from a two frame point based motion algorithm the optimal estimator turns out to be nonlinear and for that reason techniques that provide very good initial guesses for the iterative computation of the optimal estimator are developed

frobenius and cartier algebras of stanley reisner rings we study the generation of the frobenius algebra of the injective hull of a complete stanley reisner ring over a field with positive characteristic in particular by extending the ideas used by m katzman to give a counterexample to a question raised by g lyubeznik and k e smith about the finite generation of frobenius algebras we prove that the frobenius algebra of the injective hull of a complete stanley reisner ring can be only principally generated or infinitely generated also by using our explicit description of the generators of such algebra and applying the recent work by m blickle about cartier algebras and generalized test ideals we are able to show that the set of f jumping numbers of generalized test ideals associated to complete stanley reisner rings form a discrete subset inside the non negative real numbers

behavior of test ideals under smooth and etale homomorphisms we investigate the behavior of the test ideal of an excellent reduced ring of prime characteristic under base change it is shown that if h a d is a smooth homomorphism then ad d assuming that all residue fields of a at maximal ideals are perfect and that formation of the test ideal commutes with localization it is also shown that if h a m d is a finite flat homomorphism of gorenstein normal rings etale in codimension 1 then ad d more generally this last result holds under the assumption that the closed fiber of h a m d is gorenstein provided one knows that the tight closure of zero and the finitistic tight closure of zero in the injective hulls of the residue fields of a and s are equal

f injective singularities are du bois in this paper we prove that singularities of f injective type are du bois this extends the correspondence between singularities associated to the minimal model program and singularities defined by the action of frobenius in positive characteristic

convection patterns in large aspect ratio systems abstract a theory which should have widespread application is developed to treat the statics and slow dynamics of patterns of convective rolls encountered in large aspect ratio rayleigh benard boxes for case of presentation the theory is developed using model equations wavenumber selection the shape of patterns stability and the time dependence resulting from long wavelength instabilities are discussed the effects of adding the local mean drift important in low prandtl number situations are investigated our theory includes the notion of the busse stability balloon reduces near critical values of the stress parameter to the newell whitehead segel equations and contains the pomeau manneville phase equation it also gives a detailed description of the way in which the field amplitude which is slaved to the phase gradient away from threshold becomes an independent order parameter near the critical point

uniqueness and estimation of three dimensional motion parameters of rigid objects with curved surfaces two main results are established in this paper first we show that seven point correspondences are sufficient to uniquely determine from two perspective views the three dimensional motion parameters within a scale factor for the translations of a rigid object with curved surfaces the seven points should not be traversed by two planes with one plane containing the origin nor by a cone containing the origin second a set of essential parameters are introduced which uniquely determine the motion parameters up to a scale factor for the translations and can be estimated by solving a set of linear equations which are derived from the correspondences of eight image points the actual motion parameters can subsequently be determined by computing the singular value decomposition svd of a 3 3 matrix containing the essential parameters

comparison of symbolic and ordinary powers of ideals all given rings in this paper are commutative associative with identity and noetherian recently l ein r lazarsfeld and k smith els discovered a remarkable and surprising fact about the behavior of symbolic powers of ideals in affine regular rings of equal characteristic 0 if h is the largest height of an associated prime of i then i hn i n for all n 0 here if w is the complement of the union of the associated primes of i i t denotes the contraction of i t rw to r where rw is the localization of r at the multiplicative system w their proof depends on the theory of multiplier ideals including an asymptotic version and in particular requires resolution of singularities as well as vanishing theorems we want to acknowledge that without their generosity and quickness in sharing their research this manuscript would not exist our objective here is to give stronger results that can be proved by methods that are in some ways more elementary our results are valid in both equal characteristic 0 and in positive prime characteristic p but depend on reduction to characteristic p we use tight closure methods and in consequence we need neither resolution of singularities nor vanishing theorems that may fail in positive characteristic for the most basic form of the result all that we need from tight closure theory is the definition of tight closure and the fact that in a regular ring every ideal is tightly closed we note that the main argument here is closely related to a proof given in hu 5 14 16 p 45 that regular local rings in characteristic p are ufds

a generalization of tight closure and multiplier ideals we introduce a new variant of tight closure associated to any fixed ideal a which we call tight closure and study various properties thereof in our theory the annihilator ideal r a of all tight closure relations which is a generalization of the test ideal in the usual tight closure theory plays a particularly important role we prove the correspondence of the ideal a and the multiplier ideal associated to a or the adjoint of a in lipman s sense in normal q gorenstein rings reduced from characteristic zero to characteristic p 0 also in fixed prime characteristic we establish some properties of r a similar to those of multiplier ideals e g a briancon skoda type theorem subadditivity etc with considerably simple proofs and study the relationship between the ideal and the f rationality of rees algebras

frobenius splitting and cohomology vanishing for schubert varieties let k be an algebraically closed field of characteristic p 0 and x be a projective variety over k we then have the absolute frobenius f x x and an injection ox f 6x given by f fp f e o9x if this makes ox a direct summand in f ox as an x module we call x a frobenius split variety for such a variety the vanishing theorem for ample line bundles follows trivially from serre s vanishing theorem for tensoring ox f ox by an ample line bundle l and noting that l f x f f l f lp projection formula we get that the map in cohomology h x l h x f lp h x lp is an injection iterating this we see that h x l injects into h x lp for every p but for large v the latter is zero thus frobenius split varieties have quite pleasant properties it also turns out that using duality for the frobenius morphism or equivalently the cartier operator one can give a manageable criterion for frobenius splitting the point here is the local nature of duality the relevance of the compatibility of local and global duality was suggested to us by grothendieck s proof of h x y 0 for a noncomplete variety x of dimension n 4 theorem 6 9 and by kempf s paper 10 by this criterion and the bott samelson demazure desingularisation of schubert varieties it follows very easily that schubert varieties in characteristic p are frobenius split the following vanishing theorem is then an immediate consequence let g be a reductive group over the field k of arbitrary characteristic zero or positive let q be a parabolic subgroup and x c g q a schubert variety let l be an ample line bundle on g q then hi x l 0 for i 0 and the restriction map h0 g q l h0 x l is surjective if char k 0 this is a consequence of the compatible frobenius splitting of x in g q and the char k 0 case is handled by semicontinuity the above result for special schubert varieties has been proved by seshadrimusili lakshmibai 12 13 and by kempf 9 by using characteristic free methods for x g q andersen 1 and haboush 5 have given simple proofs using characteristic p methods

test ideals in local rings it is shown that certain aspects of the theory of tight closure are well behaved under localization let j be the parameter test ideal for r a complete local cohen macaulay ring of positive prime characteristic for any multiplicative system u c r it is shown that ju 1r is the parameter test ideal for u 1 r this is proved by proving more general localization results for the here introduced classes of f ideals of r and f submodules of the canonical module of r which are annihilators of r modules with an action of frobenius it also follows that the parameter test ideal cannot be contained in any parameter ideal of r tight closure has produced surprising new results for instance that the absolute integral closure of a complete local domain r of prime characteristic is a cohen macaulay algebra for r as well as tremendously simple proofs for otherwise difficult theorems such as the fact that the ring of invariants of a linearly reductive group acting on a regular ring is cohen macaulay the definition of tight closure is recalled in section 1 but we refer the reader to the papers of hochster and huneke listed in the bibliography for more about its applications while primarily a prime characteristic notion tight closure offers insight into arbitrary commutative rings containing q by fairly standard reduction to characteristic p techniques despite its successes the tight closure operation which in its principal setting is a closure operation performed on ideals in a commutative noetherian ring of prime characteristic remains poorly understood for example tight closure is not known to behave well under localization given an ideal i of a ring r letting i denote the tight closure of i it is not known whether or not

drops in joint hilbert kunz multiplicities and projective equivalence abstract given two m primary ideals a and i in a noetherian local or graded ring we introduce the level ideals of i with respect to a as a measure of the asymptotic growth of the drop in the joint hilbert kunz multiplicity e hk a t i e hk a t i x we study the properties that must hold for two ideals a b to determine the same level ideals for all i and we show that this is related to projective equivalence

instability of flows in spatially developing media this paper studies specific forms of partial differential equations which represent the amplitude evolution of disturbances to flows in media whose properties vary slowly in space over a length scale epsilon 1 a method is described for finding series solutions in the small parameter epsilon and general results for the green s function and the time periodic response are obtained with the aid of a computer based symbolic algebra manipulation package the method permits results to be obtained to any desired accuracy for any specified spatial variation the method is then applied to find the green s function for the linearized ginzburg landau equation which is a good simple model of instability growth in many marginally unstable fluid dynamical systems in the case where the instability parameter mu normally a constant in the equation varies with the spatial coordinate even though the method produces only power series solutions in epsilon it is shown that with its help exact analytic solutions valid for all epsilon however large can be found in some cases the exact solutions for the cases of linear and quadratic spatial variation of mu are obtained and discussed and criteria are found for the onset of global instability comparisons are made between these criteria and the local absolute and convective instability criteria and the results are found to be in qualitative agreement with other authors numerical and analytic results in particular it is shown that a small region of local absolute instability up to a maximal size which is known analytically may exist and yet that the flow may be globally stable it is also shown that in the case of quadratic spatial variation of mu with an infinite region of local absolute instability the green s function may for some particular parameter ranges become infinite at all spatial locations simultaneously at some finite time

f regularity does not deform we show that the property of f regularity does not deform and thereby settle a long standing open question in the theory of tight closure specifically we construct a three dimensional graded domain r which is not f regular or even f pure but has a quotient r tr which is f regular examples are constructed over fields of characteristic p 0 as well as over fields of characteristic zero 1 introduction throughout this paper all rings are commutative noethe rian and have an identity element the theory of tight closure was developed by melvin hochster and craig huneke in hh2 and draws attention to rings which have the property that all their ideals are tightly closed called weakly f regular rings the term f regular is reserved for rings all of whose localizations are weakly f regular a natural question that arose with the development of the the ory was whether the property of f regularity deforms i e if r m k is a local ring such that r tr is f regular for some nonzerodivisor t m m ustr be f regular see the epilogue of ho hochster and huneke showed that this is indeed true if the ring r is gorenstein hh3 and their work has been followed by various attempts at extending this result see akm si sm3 our pri mary goal here is to settle this question by constructing a family of examples to show that f regularity does not deform we shall throughout be considering graded rings but local examples can be obtained in all cases by localizing at the homogeneous maximal ideals our main result is theorem 1 1 there exists an graded ring r of dimension three finitely generated over a field r 0 k of characteristic p 2 which is not f pure but has an f regular quotient r tr where t m is a homogeneous nonzerodivisor specifically for positive integers m and n satisfying m m n 2 consider the ring r k a b c d t i where i is generated by the size two minors of the

some experiments on estimating the 3 d motion parameters of a rigid body from two consecutive image frames this paper describes the results of some experiments on estimating the 3 d motion parameters of a rigid body from two consecutive tv images and discusses several factors which affect the accuracy of the results these factors include the sizes of the motion parameters themselves the accuracy of the raw data and the number of point correspondences in addition we address two related topics determining corner positions to subpixel accuracy and matching point patterns with different scales

observability of 3d motion this paper examines the inherent difficulties in observing 3d rigid motion from image sequences it does so without considering a particular estimator instead it presents a statistical analysis of all the possible computational models which can be used for estimating 3d motion from an image sequence these computational models are classified according to the mathematical constraints that they employ and the characteristics of the imaging sensor restricted field of view and full field of view regarding the mathematical constraints there exist two principles relating a sequence of images taken by a moving camera one is the epipolar constraint applied to motion fields and the other the positive depth constraint applied to normal flow fields 3d motion estimation amounts to optimizing these constraints over the image a statistical modeling of these constraints leads to functions which are studied with regard to their topographic structure specifically as regards the errors in the 3d motion parameters at the places representing the minima of the functions for conventional video cameras possessing a restricted field of view the analysis shows that for algorithms in both classes which estimate all motion parameters simultaneously the obtained solution has an error such that the projections of the translational and rotational errors on the image plane are perpendicular to each other furthermore the estimated projection of the translation on the image lies on a line through the origin and the projection of the real translation the situation is different for a camera with a full 360 degree field of view achieved by a panoramic sensor or by a system of conventional cameras in this case at the locations of the minima of the above two functions either the translational or the rotational error becomes zero while in the case of a restricted field of view both errors are non zero although some ambiguities still remain in the full field of view case the implication is that visual navigation tasks such as visual servoing involving 3d motion estimation are easier to solve by employing panoramic vision also the analysis makes it possible to compare properties of algorithms that first estimate the translation and on the basis of the translational result estimate the rotation algorithms that do the opposite and algorithms that estimate all motion parameters simultaneously thus providing a sound framework for the observability of 3d motion finally the introduced framework points to new avenues for studying the stability of image based servoing schemes

f thresholds of hypersurfaces for every 0 that is f finite i e such that the frobenius morphism f r r is finite if a is an ideal in r and if is a nonnegative real number then the corresponding test ideal is denoted by a in this context we say that is an f jumping exponent or an f threshold if a 6 a for every the following is our main result about f jumping exponents in positive characteristic theorem 1 1 if r is an f finite regular ring and if a f is a principal ideal then the f jumping exponents of a are rational and they form a discrete set the discreteness and the rationality of f jumping numbers has been proved in bms for every ideal when the ring r is essentially of finite type over an f finite field we mention also that for r k x y with k a finite field the above result has been proved in ha using a completely different approach we stress that the difficulty in attacking this result does notcome from the fact that there is no available resolution of singularities in positive characteristic even

uniform bounds and symbolic powers on smooth varieties the purpose of this note is to show how one can use multiplier ideals toestablish effective uniform bounds on the multiplicative behavior of certainfamilies of ideal sheaves on a smooth algebraic variety in particular weprove a quick but rather surprising result concerning the symbolic powersof radical ideals on such a variety let

on the discreteness and rationality of f jumping coefficients abstract we prove that the f jumping coefficients of a principal ideal of an excellent regular local ring of characteristic p 0 are all rational and form a discrete subset of r

discreteness and rationality of f jumping numbers on singular varieties we prove that the f jumping numbers of the test ideal tau x delta mathfrak a t are discrete and rational under the assumptions that x is a normal and f finite scheme over a field of positive characteristic p kx is mathbb q cartier of index not divisible p and either x is essentially of finite type over a field or the sheaf of ideals mathfrak a is locally principal this is the largest generality for which discreteness and rationality are known for the jumping numbers of multiplier ideals in characteristic zero

weak f regularity deforms in q gorenstein rings weakly f regular inother words does the property of weak f regularity deform it has beenshown by singh that in general weak f regularity does not deform si2 wxit is known that f rationality deforms hh2 4 2 and the proof iswxstraightforward however the same difficulties that arise when trying to

centers of f purity in this paper we study a positive characteristic analogue of the centers of log canonicity of a pair r we call these analogues centers of f purity we prove positive characteristic analogues of subadjunction like results prove new stronger subadjunction like results and in some cases lift these new results to characteristic zero using a generalization of centers of f purity which we call uniformly f compatible ideals we give a characterization of the test ideal which unifies several previous characterizations finally in the case that 0 we show that uniformly f compatible ideals coincide with the annihilators of the mathcal f e r k submodules of e r k as defined by lyubeznik and smith

formulas for multiplier ideals on singular varieties we prove a generalization of demailly ein lazarsfeld s subadditivity formula and musta s summation formula for multiplier ideals to the case of singular varieties using characteristic p methods as an application of our formula we improve hochster huneke s result on the growth of symbolic powers of ideals in singular affine algebras

f signature of pairs and the asymptotic behavior of frobenius splittings abstract we generalize f signature to pairs r d where d is a cartier subalgebra on r as defined by the first two authors in particular we show the existence and positivity of the f signature for any strongly f regular pair in one application we answer an open question of aberbach and enescu by showing that the f splitting ratio of an arbitrary f pure local ring is strictly positive furthermore we derive effective methods for computing the f signature and the f splitting ratio in the spirit of the work of r fedder

the fundamental matrix theory algorithms and stability analysis in this paper we analyze in some detail the geometry of a pair of cameras i e a stereo rig contrarily to what has been done in the past and is still done currently for example in stereo or motion analysis we do not assume that the intrinsic parameters of the cameras are known coordinates of the principal points pixels aspect ratio and focal lengths this is important for two reasons first it is more realistic in applications where these parameters may vary according to the task active vision second the general case considered here captures all the relevant information that is necessary for establishing correspondences between two pairs of images this information is fundamentally projective and is hidden in a confusing manner in the commonlyused formalism of the essential matrix introduced by longuet higgins 1981 this paper clarifies the projective nature of the correspondence problem in stereo and shows that the epipolar geometry can be summarized in one 3 x 3 matrix of rank 2 which we propose to call the fundamental matrix after this theoretical analysis we embark on the task of estimating the fundamental matrix from point corre spondences a task which is of practical importance we analyze theoretically and compare experimentally using synthetic and real data several methods of estimation the problem of the stability of the estimation is studied from two complementary viewpoints first we show that there is an interesting relationship between the fundamental matrix and three dimensional planes which induce homographies between the images and create unstabilities in the estimation procedures second we point to a deep relation between the unstability of the estimation procedure and the presence in the scene of so called critical surfaces which have been studied in the context of motion analysis finally we conclude by stressing the fact that we believe that the fundamental matrix will play a crucial role in future applications of three dimensional computer vision by greatly increasing its versatility robustness and hence applicability to real difficult problems

3 d motion estimation understanding and prediction from noisy image sequences this paper presents an approach to understanding general 3 d motion of a rigid body from image sequences based on dynamics a locally constant angular momentum lcam model is introduced the model is local in the sense that it is applied to a limited number of image frames at a time specifically the model constrains the motion over a local frame subsequence to be a superposition of precession and translation thus the instantaneous rotation axis of the object is allowed to change through the subsequence the trajectory of the rotation center is approximated by a vector polynomial the parameters of the model evolve in time so that they can adapt to long term changes in motion characteristics the nature and parameters of short term motion can be estimated continuously with the goal of understanding motion through the image sequence the estimation algorithm presented in this paper is linear i e the algorithm consists of solving simultaneous linear equations based on the assumption that the motion is smooth object positions and motion in the near future can be predicted and short missing subsequences can be recovered noise smoothing is achieved by overdetermination and a leastsquares criterion the framework is flexible in the sense that it allows both overdetermination in number of feature points and the number of image frames

unbiased estimation and statistical analysis of 3 d rigid motion from two views the problem of estimating 3d rigid motion from point correspondences over two views is formulated as nonlinear least squares ls optimization and the statistical behaviors of the errors in the solution are analyzed by introducing a realistic model of noise described in terms of the covariance matrices of n vectors it is shown that the ls solution based on the epipolar constraint is statistically biased the geometry of this bias is described in both quantitative and qualitative terms finally an unbiased estimation scheme is presented and random number simulations are conducted to observe its effectiveness

estimation three dimensional motion of rigid objects from noisy observations an estimate is made of the motion of a rigid body from two noisy 2 d perspective projections using the least squares method and the algebra of r y tsai and t s huang 1984 the accuracy of the estimated motion parameters is influenced by the position of the features of the object used in the calculation four test variables are derived that indicate how the accuracy is affected and they are used for discarding inaccurate estimates monte carlo tests demonstrate the obtained accuracy

uniform approximation of abhyankar valuation ideals in smooth function fields fix a rank one valuation centered at a smooth point x on an algebraic variety over a field of characteristic zero assume that is abhyankar that is that its rational rank plus its transcendence degree equal the dimension of the variety let a m denote the ideal of elements in the local ring of x whose valuations are at least m our main theorem is that there exists k 0 such that a mn is contained in a m k n for all m and n this can be viewed as a greatly strengthened form of izumi s theorem for abhyankar valuations centered on smooth complex varieties the proof uses the theory of asymptotic multiplier ideals

adjoint ideals along closed subvarieties of higher codimension in this paper we introduce a notion of adjoint ideal sheaves along closed subvarieties of higher codimension and study its local properties using char acteristic p methods when x is a normal gorenstein closed subvariety of a smooth complex variety a we formulate a restriction property of the adjoint ideal sheaf adj x a of a along x involving the l c i ideal sheaf dx of x the proof relies on a modification of generalized test ideals of hara and yoshida 11

f singularities of pairs and inversion of adjunction of arbitrary codimension we generalize the notions of f regular and f pure rings to pairs r mathfrak a t of rings r and ideals mathfrak a subset r with real exponent t 0 and investigate these properties these f singularities of pairs correspond to singularities of pairs of arbitrary codimension in birational geometry via this correspondence we prove a sort of inversion of adjunction of arbitrary codimension which states that for a pair x y of a smooth variety x and a closed subscheme y subsetneq x if the restriction z y z to a normal gorenstein closed subvariety z subsetneq x is klt resp lc then the pair x y z is plt resp lc near z

castelnuovo mumford regularity and the discreteness of f jumping coefficients in graded rings in this paper we show that the sets of f jumping coefficients of ideals form discrete sets in certain graded f finite rings we do so by giving a criterion based on linear bounds for the growth of the castelnuovo mumford regularity of certain ideals we further show that these linear bounds exists for one dimensional rings and for ideals of most two dimensional domains we conclude by applying our technique to prove that all sets of f jumping coefficients of all ideals in the determinantal ring given as t he quotient by 2 2 minors in a 2 3 matrix of indeterminates form discrete sets

on the constancy regions for mixed test ideals abstract in this note we study the partition of r 0 n given by the regions where the mixed test ideals a 1 t 1 a n t n are constant we show that each region can be described as the preimage of a natural number under a p fractal function r 0 n n in addition we give some examples illustrating that these regions do not need to be composed of finitely many rational polytopes

the f pure threshold of a calabi yau hypersurface we compute the f pure threshold of the affine cone over a calabi yau hypersurface and relate it to the order of vanishing of the hasse invariant on the versal deformation space of the hypersurface

multiplier ideal sheaves and kahler einstein metrics of positive scalar curvature we present a method for proving the existence of kiihler einstein metrics of positive scalar curvature on certain compact complex manifolds and use the method to produce a large class of examples of compact kiihler einstein manifolds of positive scalar curvature suppose that m is a compact complex manifold of positive first chern class as is well known the existence of a kiihler einstein metric on m is equivalent to the existence of a solution to a certain complex monge ampere equation on m to solve this complex mongeampere equation by the method of continuity one needs only to establish the appropriate zeroth order a priori estimate suppose now that m does not admit a kiihler einstein metric so that the zeroth order a priori estimate fails to hold from this lack of an estimate we extract various global algebro geometric properties of m by introducing a coherent sheaf of ideals j on m called the multiplier ideal sheaf which carefully measures the extent to which the estimate fails the sheaf y is analogous to the subelliptic multiplier ideal sheaf that j j kohn introduced over a decade ago to obtain sufficient conditions for subellipticity of the d neumann problem now j is a global algebro geometric object on m and it so happens that j satisfies a number of highly nontrivial global algebro geometric conditions including a cohomology vanishing theorem in particular the complex analytic subspace v c m cut out by j is nonempty connected and has arithmetic genus zero if v is zero dimensional then it is a single reduced point while if v is one dimensional then its support is a tree of smooth rational curves the logarithmic geometric genus of m v always vanishes these considerations place nontrivial global algebro geometric restric

cartier modules finiteness results on a locally noetherian scheme x over a field of positive characteristic p we study the category of coherent ox modules m equipped with a pe linear map i e an additive map c ox ox satisfying rc m c r p e m for all m a m r a ox the notion of nilpotence meaning that some power of the map c is zero is used to rigidify this cate gory the resulting quotient category called cartier crystals satisfies some strong finiteness conditions the main result in this paper states that if the frobenius morphism on x is a finite map i e if x is f finite then all cartier crystals have finite length we further show how this and related results can be used to recover and generalize other finiteness results of hartshorne speiser 19 lyubeznik 27 sharp 30 enescu hochster 14 and hochster 20 about the structure of modules with a left action of the frobenius for example we show that over any regular f finite scheme x lyubeznik s f finite modules 27 have finite length

globalizing f invariants abstract in this paper we define and study the global hilbert kunz multiplicity and the global f signature of prime characteristic rings which are not necessarily local our techniques are made meaningful by extending many known theorems about hilbert kunz multiplicity and f signature to the non local case

least squares fitting of two 3 d point sets two point sets pi and p i i 1 2 n are related by p i rpi t ni where r is a rotation matrix t a translation vector and ni a noise vector given pi and p i we present an algorithm for finding the least squares solution of r and t which is based on the singular value decomposition svd of a 3 3 matrix this new algorithm is compared to two earlier algorithms with respect to computer time requirements

fully nonlinear global modes in spatially developing media abstract global modes on a doubly infinite one dimensional domain wkbj approximations in the linear and nonlinear regions through appropriate transition layers the real global frequency is determined by requiring that spatial branches issuing from x and x be continuously connected at a saddle point of the local nonlinear dispersion relation n l k r 2 x between the frequency the wave number k and amplitude r at a given station x the results constitute a fully nonlinear generalization of the linear frequency selection criteria previously obtained by chomaz et al 1991 monkewitz et al 1993 and le dizes et al 1996

when does the subadditivity theorem for multiplier ideals hold demailly ein and lazarsfeld proved the subadditivity theorem for multiplier ideals on nonsingular varieties which states the multiplier ideal of the product of ideals is contained in the product of the individual multiplier ideals we prove that in the two dimensional case the subadditivity theorem holds on log terminal singularities however in the higher dimensional case we have several counterexamples we consider the subadditivity theorem for monomial ideals on toric rings and construct a counterexample on a three dimensional toric ring

on f pure thresholds using the frobenius map we introduce a new invariant for a pair r a of a ring r and an ideal a r which we call the f pure threshold c a of a and study its properties we see that the f pure threshold characterizes several ring theoretic properties by virtue of hara and yoshida s result trans amer math soc 355 2003 3143 the f pure threshold c a in characteristic zero corresponds to the log canonical threshold lc a which is an important invariant in birational geometry using the f pure threshold we prove some ring theoretic properties of three dimensional terminal singularities of characteristic zero also in fixed prime characteristic we establish several properties of the f pure threshold similar to those of the log canonical threshold with quite simple proofs

d modules bernstein sato polynomials and f invariants of direct summands abstract we study the structure of d modules over a ring r which is a direct summand of a polynomial or a power series ring s with coefficients over a field we relate properties of d modules over r to d modules over s we show that the localization r f and the local cohomology module h i i r have finite length as d modules over r furthermore we show the existence of the bernstein sato polynomial for elements in r in positive characteristic we use this relation between d modules over r and s to show that the set of f jumping numbers of an ideal i r is contained in the set of f jumping numbers of its extension in s as a consequence the f jumping numbers of i in r form a discrete set of rational numbers we also relate the bernstein sato polynomial in r with the f thresholds and the f jumping numbers in r

slowly varying fully nonlinear wavetrains in the ginzburg landau equation abstract following the ideas of howard and kopell 9 a perturbation theory is developed for slowly varying fully nonlinear wavetrains i e solutions which appear locally as travelling waves but with frequencies and wavelengths which may vary widely on long length and time scales this perturbation theory is applied to the ginzburg landau equation the motion and stability of slowly varying wavetrains is shown to be governed by a simple wave equation which can develop shocks corresponding to rapid changes in wavenumber numerical results supporting this theory are presented a shock structure is proposed and numerically verified these results together with a winding invariant valid in the limit of slow variation suggest that over a large range of parameters many initial conditions relax to uniform wavetrains the evolution of a marginally diffusively stable wavetrain is also examined it is argued that the evolution is governed by a perturbed korteweg de vries equation

bifurcation to fully nonlinear synchronized structures in slowly varying media the selection of fully nonlinear extended oscillating states is analyzed in the context of one dimensional nonlinear evolution equations with slowly spatially varying coefficients on a doubly infinite domain two types of synchronized structures referred to as steep and soft global modes are shown to exist steep global modes are characterized by the presence of a sharp stationary front at a marginally absolutely unstable station and their frequency is determined by the corresponding linear absolute frequency as in dee langer propagating fronts soft global modes exhibit slowly varying amplitude and wave number over the entire domain and their frequency is determined by the application of a saddle point condition to the local nonlinear dispersion relation the two selection criteria are compared and shown to be mutually exclusive the onset of global instability first gives rise to a steep global mode via a saddle node bifurcation as soon as local linear absolute instability is reached somewhere in the medium as a result such self sustained structures may be observed while the medium is still globally stable in a strictly linear approximation soft global modes only occur further above global onset and for sufficiently weak advection the entire bifurcation scenario and state diagram are described in terms of three characteristic control parameters the complete spatial structure of nonlinear global modes is analytically obtained in the framework of wkbj approximations 2001 elsevier science b v all rights reserved

depths of multiplier ideals and integral closure in this note we study how the depths of multiplier ideals behave under restriction we also study possible values of the depths of multiplier ideals in the filtrations induced from maximal ideal sheaves we then use it to give a sufficient condition for the integral closedness of the product of a multiplier ideal and a power of maximal ideal sheaf in the spirit of huneke

f signature exists suppose r is a d dimensional reduced f finite noetherian local ring with prime characteristic p 0 and perfect residue field let r 1 p e be the ring of p e th roots of elements of r for e and let a e denote the maximal rank of a free r module appearing in a direct sum decomposition of r 1 p e we show the existence of the limit s r lim e to infty frac a e p ed called the f signature of r this invariant which can be extended to all local rings in prime characteristic was first formally defined by c huneke and g leuschke in math ann 324 2 391 404 2002 and has previously been shown to exist only in special cases the proof of our main result is based on the development of certain uniform hilbert kunz estimates of independent interest additionally we analyze the behavior of the f signature under finite ring extensions and recover explicit formulae for the f signatures of arbitrary finite quotient singularities

rees algebras of f regular type abstract we study the f regularity of rees algebras r i a it in terms of the global f regularity of the blowing up x proj r i of spec a as it reads global f regularity is a global analog of strong f regularity defined via splitting of frobenius maps in prime characteristic and these notions are extended to characteristic zero by reduction modulo p 0 we study in detail the case where a m is a two dimensional local ring and i is an m primary ideal in characteristic zero the condition for r i to have f regular type is described in terms of the dual graph of a resolution x on which i o x is invertible we also prove some miscellaneous results concerning singularities of rees algebras and extended rees algebras of higher dimension

on the existence of f thresholds and related limits we show the existence of f thresholds in full generality in addition we study properties of standard graded algebras over a field for which f pure threshold and f threshold at the irrelevant maxi

test multiplier and invariant ideals abstract this paper gives an explicit formula for the multiplier ideals and consequently for the log canonical thresholds of any gl v gl w invariant ideal in s sym v w where v and w are vector spaces over a field of characteristic 0 this characterization is done in terms of a polytope constructed from the set of young diagrams corresponding to the schur modules generating the ideal our approach consists in computing the test ideals of some invariant ideals of s in positive characteristic namely we compute the test ideals and so the f pure thresholds of any sum of products of determinantal ideals not all the invariant ideals are as the latter not even in characteristic 0 but they are up to integral closure and this is enough to reach our goals the results concerning the test ideals are obtained as a consequence of general results holding true in a special situation within such framework fall determinantal objects of a generic matrix as well as of a symmetric matrix and of a skew symmetric one similar results are thus deduced for the gl v invariant ideals in sym sym 2 v and in sym 2 v monomial ideals also fall in this framework thus we recover howald s formula for their multiplier ideals and more generally hara yoshida s formula for their test ideals in the proof we introduce the notion of floating test ideals a property that in a sense is satisfied by ideals defining schemes with the nicest possible singularities as will be shown products of determinantal ideals and by passing to characteristic 0 ideals generated by a single schur module have this property

absolute and convective instabilities front velocities and global modes in nonlinear systems abstract we study the existence of self sustained saturated solutions of the real ginzburg landau equation subject to a boundary condition at x 0 such solutions are called nonlinear global ng modes the ng instability referring to the existence of these solutions is rigorously determined and the scaling behavior of the ng modes close to threshold is derived the ng instability is first compared to the linear concept of convective absolute c a instability characterizing whether the impulse response of an unstable flow in an infinite domain is asymptotically damped or amplified at a fixed location ng modes are shown to exist while at the same time the flow may be linearly stable convectively unstable or absolutely unstable the growth size of the ng modes is shown to be proportional to 1 2 when ng and a instabilities exist simultaneously being the criticality parameter whereas a ln 1 scaling is found when the ng instability occurs while the flow is c unstable or linearly stable the nonlinear convective absolute nc na instability defined chomaz 1992 by considering in infinite homogeneous domains whether the front separating a bifurcated state from the basic state moves downstream or upstream is determined using van saarloos and hohenberg 1992 results for the selected front velocity remarkably the na domain and the ng domain are shown to coincide similar results are presented for supercritical bifurcating systems for the van der pol duffing system and for a transcritical model in all the cases the a instability is only a sufficient condition for the existence of an ng mode and these simple models demonstrate that a system may be nonlinearly absolutely unstable whereas it is linearly convectively unstable this property should be generic if one accepts the conjecture that the selected front velocity is always larger than the linear front velocity response to a constant forcing applied at the origin is also studied it is shown that in the ng region the system possesses intrinsic dynamics which cannot be removed by the forcing by contrast the behavior of a nonlinear spatial amplifier is observed in a domain larger than the nc region nc instability is only a sufficient condition to trigger the system with forcing

f rational rings have rational singularities it is proved that an excellent local ring of prime characteristic in which a single ideal generated by any system of parameters is tightly closed must be pseudorational a key point in the proof is a characterization of f rational local rings as those cohen macaulay local rings r m in which the local cohomology module h d m r where d is the dimension of r have no submodules stable under the natural action of the frobenius map an analog for finitely generated algebras over a field of characteristic zero is developed which yields a reasonably checkable tight closure test for rational singularities of an algebraic variety over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i without reference to a desingularization

big tight closure test elements for some non reduced excellent rings abstract this paper is concerned with existence of big tight closure test elements for a commutative noetherian ring r of prime characteristic p let r denote the complement in r of the union of the minimal prime ideals of r a big test element for r is an element of r which can be used in every tight closure membership test for every r module and not just the finitely generated ones the main results of the paper are that if r is excellent and satisfies condition r 0 and c r is such that r c is gorenstein and weakly f regular then some power of c is a big test element for r if i r is a homomorphic image of an excellent regular ring of characteristic p for which the frobenius homomorphism is intersection flat or ii r is f pure or iii r is local the gamma construction is not used

robust estimation of three dimensional motion parameters from a sequence of image frames using regularization in this paper we look at the issue of accurate estimation of the three dimensional motion parameters of a rigid body from a sequence of synthetic images and relate the effect of some parameters to the shape of an error function we first consider the case where only a small set of corresponding points is identified and suggest that a technique called regularization improves the quality and stability of a solution we then observe that if more pairs of corresponding points are available the error function becomes smooth and the solution stable finally we try to improve the quality of estimation by considering more than two consecutive frames for a moving camera looking at a stationary scene and summing the error functions obtained for any two consecutive frames surprisingly enough this technique does not improve stability unless we use regularization again

bernstein sato polynomials in positive characteristic abstract in characteristic zero the bernstein sato polynomial of a hypersurface can be described as the minimal polynomial of the action of an euler operator on a suitable d module we consider analogous d modules in positive characteristic and use them to define a sequence of bernstein sato polynomials corresponding to the fact that we need to consider also divided powers euler operators we show that the information contained in these polynomials is equivalent to that given by the f jumping exponents of the hypersurface in the sense of hara and yoshida n hara k i yoshida a generalization of tight closure and multiplier ideals trans amer math soc 355 2003 3143 3174

f rationality of rees algebras in this paper using the notion of the tight integral closure we will give a criterion for f rationality of rees algebras of m primary ideals in a cohen macaulay local ring as its application we prove the following results 1 in dimension two if a is f rational and i is integrally closed then the rees algebra r i is f rational on the other hand in higher dimensions we construct many examples of cohen macaulay normal rees algebras which are not f rational 2 if both a and r i are f rational then so is the extended rees algebra r i 3 if r i is f rational and a g i 1 then a is f rational on the other hand using resolution of singularities we will prove that a two dimensional rational singularity always admits f rational rees algebras in particular this theorem gives another way than that devised by watanabe 1997 j pure appl algebra122 323 328 to construct counterexamples to the boutot type theorem for f rational rings