learning eigenfunctions links spectral embedding and kernel pca in this letter we show a direct relation between spectral embedding methods and kernel principal components analysis and how both are special cases of a more general learning problem learning the principal eigenfunctions of an operator defined from a kernel and the unknown data generating density whereas spectral embedding methods provided only coordinates for the training points the analysis justifies a simple extension to out of sample examples the nystrom formula for multidimensional scaling mds spectral clustering laplacian eigenmaps locally linear embedding lle and isomap the analysis provides for all such spectral embedding methods the definition of a loss function whose empirical average is minimized by the traditional algorithms the asymptotic expected value of that loss defines a generalization performance and clarifies what these algorithms are trying to learn experiments with lle isomap spectral clustering and mds show that this out of sample embedding formula generalizes well with a level of error comparable to the effect of small perturbations of the training set on the embedding

the drazin inverse of updating of a square matrix with application to perturbation formula1this project is supported by national nature science foundation of china doctoral point foundation of china and youth science foundation of universities in shanghai of china 1 the jordan canonical form of a matrix is used to derive systematically the drazin inverse of modification of a square matrix in addition to perturbation formula for the drazin inverse an open problem by campbell and meyer in 1975 has partially been solved the results extend earlier work by several authors

projection methods solving rectangular systems of linear equations abstract this survey paper contains a general characterization of stationary and instationary linear iterative one step methods solving over or underdetermined linear algebraic systems techniques for convergence acceleration and some generalizations of the kaczmarz art and the de la garza methods among others the question is answered under what conditions an iteration process yields a generalized matrix inverse of the type a a 1 2 a 1 2 3 a 1 2 4 a respectively and under what conditions the resulting limit vector is a minimum norm solution a least squares solution or a solution of the initial linear system

characterizations iterative method sign pattern and perturbation analysis for the dmp inverse with its applications abstract this paper is a study on main properties and characterizations of the dmp inverse also corresponding representations and computational procedures are derived particularly an upgrade of the bordering method to the case of the dmp inverse is considered as well as applications of the dmp inverse in solving singular linear systems also the dmp inverse of an upper block triangular matrix and its sign pattern are investigated furthermore several computational procedures aimed to approximating the dmp inverse are developed implemented and tested the main of them are the limit representation a revised successive matrix squaring iterations and the gradient neural network gnn dynamical system finally some perturbation bounds and continuity of the dmp inverse are studied

global f regularity of schubert varieties with applications to d modules a projective algebraic variety x over an algebraically closed field k of positive characteristic is called globally f regular if the section ring s c 0n o h x cn of an ample line bundle c on x is strongly f regular in the sense of h chster and huneke 9 cf definition 1 1 this important notion was introduced by karen smith in 18 in this paper we prove that schubert varieties are globally f regular an im mediate consequence is that local rings of schubert varieties are strongly f regular and thereby f rational another consequence is that local rings of varieties like de terminantal varieties that can be identified with open subsets of schubert varieties cf 13 are strongly f regular let x denote a flag variety and y c x a schubert variety over k then the local cohomology sheaves hjy ox are equivariant for the action of the borel subgroup and holonomic in the sense of 4 x x niodules as an application of ir rationality of schubert varieties we apply recent results of blickle cf 2 to prove that the simple objects in the category of equivariant and holonomic px modules are precisely the local cohomology sheaves hy ox where c is the codimension of y in x using a local grothendieck cousin complex from 11 we prove that the decomposition of the local cohomology modules with support in bruhat cells is multiplicity free see 4 2 in characteristic zero the local cohomology modules with support in bruhat cells correspond to dual verma modules in this setting the decomposition behavior and the simple t x modules arise from intersection cohomology complexes of schubert varieties by the riemann hilbert correspondence if we pick the singular codi mension one schubert variety y in the full flag variety z for sl4 in characteristic zero computations in kazhdan lusztig theory show that hy oz is not a simple z module see 4 1

probabilistic dimensionality reduction via structure learning we propose an alternative probabilistic dimensionality reduction framework that can naturally integrate the generative model and the locality information of data based on this framework we present a new model which is able to learn a set of embedding points in a low dimensional space by retaining the inherent structure from high dimensional data the objective function of this new model can be equivalently interpreted as two coupled learning problems i e structure learning and the learning of projection matrix inspired by this interesting interpretation we propose another model which finds a set of embedding points that can directly form an explicit graph structure we proved that the model by learning explicit graphs generalizes the reversed graph embedding method but leads to a natural interpretation from bayesian perspective this can greatly facilitate data visualization and scientific discovery in downstream analysis extensive experiments are performed that demonstrate that the proposed framework is able to retain the inherent structure of datasets and achieve competitive quantitative results in terms of various performance evaluation criteria

some block matrices with signed drazin inverses abstract the sign pattern of a real matrix m is the 0 1 1 matrix obtained from m by replacing each entry by its sign let q m be the set of real matrices with the same sign pattern as m for any m q m if the drazin inverses of m and m have the same sign pattern then m is said to have signed drazin inverse in this paper we give a complete characterization for a class of anti triangular matrices with signed drazin inverse and present a complete characterization for sign symmetric bipartite matrices with signed drazin inverse

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

a characterization of the moore penrose inverse abstract if a is a nonsingular matrix of order n the inverse of a is the unique matrix x for which rank a i i x rank a we present a generalization of this fact to singular or rectangular matrices a to obtain an analogous result for the moore penrose inverse a of a we then give a geometric interpretation for the case that a is a positive semidefinite matrix

on a new generalized inverse for matrices of an arbitrary index the purpose of this paper is to introduce a new generalized inverse called dmp inverse associated with a square complex matrix using its drazin and moore penrose inverses dmp inverse extends the notion of core inverse introduced by baksalary and trenkler for matrices of index at most 1 in baksalary and trenkler 2010 1 to matrices of an arbitrary index dmp inverses are analyzed from both algebraic as well as geometrical approaches establishing the equivalence between them

recurrent neural network for computing outer inverse two linear recurrent neural networks for generating outer inverses with prescribed range and null space are defined each of the proposed recurrent neural networks is based on the matrix valued differential equation a generalization of dynamic equations proposed earlier for the nonsingular matrix inversion the moore penrose inversion as well as the drazin inversion under the condition of zero initial state the application of the first approach is conditioned by the properties of the spectrum of a certain matrix the second approach eliminates this drawback though at the cost of increasing the number of matrix operations the cases corresponding to the most common generalized inverses are defined the conditions that ensure stability of the proposed neural network are presented illustrative examples present the results of numerical simulations

the drazin inverse of an even order tensor and its application to singular tensor equations abstract the notion of the moore penrose inverses of matrices was recently extended from matrix space to even order tensor space with einstein product in the literature in this paper we further study the properties of even order tensors with einstein product we define the index and characterize the invertibility of an even order square tensor we also extend the notion of the drazin inverse of a square matrix to an even order square tensor an expression for the drazin inverse through the core nilpotent decomposition for a tensor of even order is obtained as an application the drazin inverse solution of the singular linear tensor equation a x b will also be included

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

probabilistic non linear principal component analysis with gaussian process latent variable models summarising a high dimensional data set with a low dimensional embedding is a standard approach for exploring its structure in this paper we provide an overview of some existing techniques for discovering such embeddings we then introduce a novel probabilistic interpretation of principal component analysis pca that we term dual probabilistic pca dppca the dppca model has the additional advantage that the linear mappings from the embedded space can easily be non linearised through gaussian processes we refer to this model as a gaussian process latent variable model gp lvm through analysis of the gp lvm objective function we relate the model to popular spectral techniques such as kernel pca and multidimensional scaling we then review a practical algorithm for gp lvms in the context of large data sets and develop it to also handle discrete valued data and missing attributes we demonstrate the model on a range of real world and artificially generated data sets

relationships between dmp relation and some partial orders let h be an infinite dimensional complex hilbert space and let b h be the algebra of all bounded linear operators on h the equivalent definition of the dmp relation on b h using operator block matrices is introduced we present a new interpretation of this relation which allows to generalize many known results for matrices to general hilbert spaces regarding the dmp relation and its relationship with the star order and minus partial order also some properties of dmp relation defined with the help of idempotent operators are investigated

a novel framework for learning geometry aware kernels the data from real world usually have nonlinear geometric structure which are often assumed to lie on or close to a low dimensional manifold in a high dimensional space how to detect this nonlinear geometric structure of the data is important for the learning algorithms recently there has been a surge of interest in utilizing kernels 1 to exploit the manifold structure of the data such kernels are called geometry aware kernels and are widely used in the machine learning algorithms the performance of these algorithms critically relies on the choice of the geometry aware kernels intuitively a good geometry aware kernel should utilize additional information other than the geometric information in many applications it is required to compute the out of sample data directly however most of the geometry aware kernel methods are restricted to the available data given beforehand with no straightforward extension for out of sample data in this paper we propose a framework for more general geometry aware kernel learning the proposed framework integrates multiple sources of information and enables us to develop flexible and effective kernel matrices then we theoretically show how the learned kernel matrices are extended to the corresponding kernel functions in which the out of sample data can be computed directly under our framework a novel family of geometry aware kernels is developed especially some existing geometry aware kernels can be viewed as instances of our framework the performance of the kernels is evaluated on dimensionality reduction classification and clustering tasks the empirical results show that our kernels significantly improve the performance 1 kernel refers to kernel function or kernel matrix which should be understood from the context

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

characterizations approximation and perturbations of the core ep inverse abstract several characterizations and representations for the core ep inverse are presented and investigated to that goal several explicit bordering technique for characterizing and defining the core ep inverse are analyzed also the matrix splitting method and the iterative successive matrix squaring algorithm for computing the core ep inverse are investigated the perturbation bounds related with the core ep inverse are also estimated

ray pattern of drazin inverses of some matrices abstract let q a be the set of complex matrices with the same ray pattern as a for any a q a if the drazin inverses of a and a have the same ray pattern then a is said to have ray drazin inverse in this paper we give some properties on matrices whose drazin inverses or m p inverses are ray unique and use these properties to obtain necessary and sufficient conditions for some block matrices to have ray drazin inverses

the global homological dimension of the ring of differential operators on a nonsingular variety over a field of positive characteristic let k be a commutative ring and a a commutative k algebra in section 2 we define d a the ring of k linear differential operators on a if k is a field of characteristic zero and a is the coordinate ring of a nonsingular affine algebraic variety over k then it is shown in 3 51 that the global homological dimension of d a gl dim d a equals the dimension of the variety here we prove that if k is an algebraically closed field of characteristic p 0 and a is the coordinate ring of a nonsingular afline algebraic variety x over k then gl dim d a dim x in s it is shown that the weak global dimension of d a w dim d a equals dim x as d a is not noetherian there is no apriori reason for w dim d a and gl dim d a to be equal it is a relatively straightforward matter to see that d a is a union of subalgebras each of which has global dimension equal to dim x and so a theorem of berstein 2 gives gl dim d a d dim x 1 so the point is to show that in this particular situation berstein s result can be improved to show gl dim d a dim x that the global dimension is bounded below by dim x is a consequence of the fact that w dim d a dim x

an efficient computation of generalized inverse of a matrix abstract we propose a hyperpower iteration for numerical computation of the outer generalized inverse of a matrix which achieves 18th order of convergence by using only seven matrix multiplications per iteration loop this yields a high efficiency index for that computational task the algorithm has a relatively mild numerical instability and we stabilize it at the price of adding two extra matrix multiplications per iteration loop this implies an efficiency index that exceeds the known record for numerically stable iterations for this task which means substantial acceleration of the long standing algorithms for an important problem of numerical linear algebra our numerical tests cover a variety of examples in the category of generalized inverses such as drazin case rectangular case and preconditioning of linear systems the test results are in good accordance with our formal study

a unifying probabilistic perspective for spectral dimensionality reduction insights and new models we introduce a new perspective on spectral dimensionality reduction which views these methods as gaussian markov random fields grfs our unifying perspective is based on the maximum entropy principle which is in turn inspired by maximum variance unfolding the resulting model which we call maximum entropy unfolding meu is a nonlinear generalization of principal component analysis we relate the model to laplacian eigenmaps and isomap we show that parameter fitting in the locally linear embedding lle is approximate maximum likelihood meu we introduce a variant of lle that performs maximum likelihood exactly acyclic lle alle we show that meu and alle are competitive with the leading spectral approaches on a robot navigation visualization and a human motion capture data set finally the maximum likelihood perspective allows us to introduce a new approach to dimensionality reduction based on l1 regularization of the gaussian random field via the graphical lasso

successive matrix squaring algorithm for computing outer inverses abstract in this paper we derive a successive matrix squaring sms algorithm to approximate an outer generalized inverse with prescribed range and null space of a given matrix a c r m n we generalize the results from the papers l chen e v krishnamurthy i macleod generalized matrix inversion and rank computation by successive matrix powering parallel computing 20 1994 297 311 y wei successive matrix squaring algorithm for computing drazin inverse appl math comput 108 2000 67 75 y wei h wu j wei successive matrix squaring algorithm for parallel computing the weighted generalized inverse a mn appl math comput 116 2000 289 296 and obtain an algorithm for computing various classes of outer generalized inverses of a instead of particular matrices used in these articles we use an appropriate matrix r c s n m s r numerical examples are presented

the d module structure of r f modules let r be a regular ring essentially of finite type over a perfect field k an r module m is called a unit r f module if it comes equipped with an isomorphism f e m m where f denotes the frobenius map on spec r and f e is the associated pullback functor it is well known that m then carries a natural d r module structure in this paper we investigate the relation between the unit r f structure and the induced d r structure on m in particular it is shown that if k is algebraically closed and m is a simple finitely generated unit r f module then it is also simple as a d r module an example showing the necessity of k being algebraically closed is also given

the intersection homology d module in finite characteristic for y a closed normal subvariety of codimension c of a smooth variety x brylinski and kashiwara showed that the local cohomology module c y x x contains a unique simple x submodule denoted by y x in this paper the analogous result is shown for x and y defined over a perfect field of finite characteristic moreover a local construction of y x is given relating it to the theory of tight closure from the construction one obtains a criterion for the x simplicity of c y x x

laplacian eigenmaps for dimensionality reduction and data representation one of the central problems in machine learning and pattern recognition is to develop appropriate representations for complex data we consider the problem of constructing a representation for data lying on a low dimensional manifold embedded in a high dimensional space drawing on the correspondence between the graph laplacian the laplace beltrami operator on the manifold and the connections to the heat equation we propose a geometrically motivated algorithm for representing the high dimensional data the algorithm provides a computationally efficient approach to nonlinear dimensionality reduction that has locality preserving properties and a natural connection to clustering some potential applications and illustrative examples are discussed

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

a family of higher order convergent iterative methods for computing the moore penrose inverse abstract in this paper we extend the iterative method for computing the inner inverse of a matrix proposed in li and li w g li z li a family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non square matrix applied mathematics and computation 215 2010 3433 3442 to compute the moore penrose inverse of a matrix and show that the generated sequence converges to the moore penrose inverse of a matrix in a higher order the performance of the method is tested on some randomly generated matrices

linear dimensionality reduction survey insights and generalizations linear dimensionality reduction methods are a cornerstone of analyzing high dimensional data due to their simple geometric interpretations and typically attractive computational properties these methods capture many data features of interest such as covariance dynamical structure correlation between data sets input output relationships and margin between data classes methods have been developed with a variety of names and motivations in many fields and perhaps as a result the connections between all these methods have not been highlighted here we survey methods from this disparate literature as optimization programs over matrix manifolds we discuss principal component analysis factor analysis linear multidimensional scaling fisher s linear discriminant analysis canonical correlations analysis maximum autocorrelation factors slow feature analysis sufficient dimensionality reduction undercomplete independent component analysis linear regression distance metric learning and more this optimization framework gives insight to some rarely discussed shortcomings of well known methods such as the suboptimality of certain eigenvector solutions modern techniques for optimization over matrix manifolds enable a generic linear dimensionality reduction solver which accepts as input data and an objective to be optimized and returns as output an optimal low dimensional projection of the data this simple optimization framework further allows straightforward generalizations and novel variants of classical methods which we demonstrate here by creating an orthogonal projection canonical correlations analysis more broadly this survey and generic solver suggest that linear dimensionality reduction can move toward becoming a blackbox objective agnostic numerical technology

matrices with doubly signed generalized inverses abstract a real matrix a is said to have a signed generalized inverse if the sign pattern of its generalized inverse a is uniquely determined by the sign pattern of a a is said to have doubly signed generalized inverse if both a and a have a signed generalized inverse matrices with doubly signed generalized inverses are generalizations of those matrices a such that both a and a 1 are s 2 ns matrices which are studied in matrices of sign solvable linear systems cambridge university press cambridge 1995 linear algebra appl 232 1996 97 in this paper we give a complete characterization of matrices with doubly signed generalized inverses we show that if a is a real matrix with no zero rows and no zero columns then a has doubly signed generalized inverses if and only if a is permutation equivalent to a direct sum of matrices of the following types a row or a column with no zero entries fully indecomposable s 2 ns matrices of order 1 or 2 matrices of the blocked form d 1 o x d 2 where d 1 and d 2 are both invertible diagonal matrices

probabilistic principal component analysis principal component analysis pca is a ubiquitous technique for data analysis and processing but one which is not based upon a probability model in this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum likelihood estimation of parameters in a latent variable model closely related to factor analysis we consider the properties of the associated likelihood function giving an em algorithm for estimating the principal subspace iteratively and discuss the advantages conveyed by the definition of a probability density function for pca

variational inference for latent variables and uncertain inputs in gaussian processes the gaussian process latent variable model gp lvm provides a flexible approach for non linear dimensionality reduction that has been widely applied however the current approach for training gp lvms is based on maximum likelihood where the latent projection variables are maximised over rather than integrated out in this paper we present a bayesian method for training gp lvms by introducing a non standard variational inference framework that allows to approximately integrate out the latent variables and subsequently train a gp lvm by maximising an analytic lower bound on the exact marginal likelihood we apply this method for learning a gp lvm from i i d observations and for learning non linear dynamical systems where the observations are temporally correlated we show that a benefit of the variational bayesian procedure is its robustness to overfitting and its ability to automatically select the dimensionality of the non linear latent space the resulting framework is generic flexible and easy to extend for other purposes such as gaussian process regression with uncertain or partially missing inputs we demonstrate our method on synthetic data and standard machine learning benchmarks as well as challenging real world datasets including high resolution video data

d module and f module length of local cohomology modules let r be a polynomial or power series ring over a field k we study the length of local cohomology modules hji r in the category of d modules and fmodules we show that the d module length of hji r is bounded by a polynomial in the degree of the generators of i in characteristic p 0 we obtain upper and lower bounds on the f module length in terms of the dimensions of frobenius stable parts and the number of special primes of local cohomology modules of r i the obtained upper bound is sharp if r i is an isolated singularity and the lower bound is sharp when r i is gorenstein and f pure we also give an example of a local cohomology module that has different d module and f module lengths

higher order convergent iterative method for computing the generalized inverse and its application to toeplitz matrices abstract the main aim of this paper is to provide a higher order convergent iterative method in order to calculate the generalized inverse of a given matrix we extend the iterative method proposed in li et al w g li z li a family of iterative methods for computing the approximate inverse of a square matrix and inner inverse of a non square matrix applied mathematics and computation 215 2010 3433 3442 to compute the 2 inverse generalized inverse a t s 2 2 3 inverse and 2 4 inverse moreover we modify the iterative method to compute the generalized inverse a t s 2 of toeplitz matrices by using the displacement theory

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

core ep decomposition and its applications abstract in this paper we introduce a new decomposition called the core ep decomposition in the present paper for square matrices and give some of its applications by applying the decomposition we derive several characterizations of the core ep inverse introduce two orders the core ep order and the core minus partial order and characterize their properties we also give some properties of the decomposition

simplicity of rings of differential operators in prime characteristic let w be a finite dimensional representation of a linearly reductive group g over a field k motivated by their work on classical rings of invariants levasseur and stafford asked whether the ring of invariants under g of the symmetric algebra of w has a simple ring of differential operators in this paper we show that this is true in prime characteristic indeed if r is a graded subring of a polynomial ring over a perfect field of characteristic p 0 and if the inclusion r hookrightarrow s splits then d k r is a simple ring in the last section of the paper we discuss how one might try to deduce the characteristic zero case from this result as yet however this is a subtle problem and the answer to the question of levasseur and stafford remains open in characteristic zero http www luc ac be research algebra 1991 mathematics subject classification 16s32 16g60 13a35

model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data we consider the problem of estimating the parameters of a gaussian or binary distribution in such a way that the resulting undirected graphical model is sparse our approach is to solve a maximum likelihood problem with an added l1 norm penalty term the problem as formulated is convex but the memory requirements and complexity of existing interior point methods are prohibitive for problems with more than tens of nodes we present two new algorithms for solving problems with at least a thousand nodes in the gaussian case our first algorithm uses block coordinate descent and can be interpreted as recursive l1 norm penalized regression our second algorithm based on nesterov s first order method yields a complexity estimate with a better dependence on problem size than existing interior point methods using a log determinant relaxation of the log partition function wainwright and jordan 2006 we show that these same algorithms can be used to solve an approximate sparse maximum likelihood problem for the binary case we test our algorithms on synthetic data as well as on gene expression and senate voting records data

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