tight double linkage of gorenstein algebras abstract a gorenstein ideal k in a local ring r is in the class if there is a sequence of linked ideals i 0 i 1 i 2 n k with i 0 a complete intersection and i 2 i gorenstein for all i if k is in then k is linked to a complete intersection by a sequence of tight double links this means there is a sequence i 0 j 0 j 1 j 2m k such that for each p j 2 p 1 is an almost complete intersection of the form b y w and its immediate neighbors are linked to it by the similar regular sequences b y and b w respectively in this sense j 2 p and j 2 p 2 are tightly linked gorenstein ideals let k be an algebraically closed field and a k x 1 x q k with k in the k algebra a is rigid if and only if it is linked to a regular complete intersection by a sequence of semi generic tight double links it follows that if a is rigid then it is regular in codimension six

syzygies of modules with positive codimension abstract a higher syzygy of a module with positive codimension is a maximal cohen macaulay module that plays an important role in cohen macaulay approximation over gorenstein rings we show that every maximal cohen macaulay module is a higher syzygy of some positive codimensional module if and only if the ring is an integral domain also we discuss the hierarchy of rings with respect to cohen macaulay approximation by codimensions of modules

transient dynamics of on line learning in two layered neural networks the dynamics of on line learning in neural networks with continuous units is dominated by plateaux in the time dependence of the generalization error using tools from statistical mechanics we show for a soft committee machine the existence of several fixed points of the dynamics of learning that give rise to complicated behaviour such as cascade like runs through different plateaux with a decreasing value of the corresponding generalization error we find learning rate dependent phenomena such as splitting and disappearing of fixed points of the equations of motion the dependence of plateau lengths on the initial conditions is described analytically and simulations confirm the results

linkage of modules abstract it is shown that the notion of linkage of algebraic varieties introduced by peskine and szpiro can be generalized to finitely generated modules over non commutative noetherian semiperfect rings besides greater generality the module theoretic approach brings about new invariants of linkage and yields improved results and simpler proofs even in the traditional settings of commutative algebraic geometry and local algebra connections with auslander reiten sequences singularity theory derived categories local cohomology buchsbaum modules and maximal cohen macaulay approximations are discussed

perfect linkage of cohen macaulay modules over cohen macaulay rings abstract in this paper we introduce and study the notion of linkage by perfect modules which we call perfect linkage for cohen macaulay modules over cohen macaulay local rings we explore perfect linkage in connection with syzygies maximal cohen macaulay approximations and yoshino isogawa linkage we recover a theorem of yoshino and isogawa and analyze the structure of double perfect linkage moreover we establish a criterion for two cohen macaulay modules of codimension one to be perfectly linked and apply it to the classical linkage theory for ideals we also construct various examples of linkage of modules and ideals

optimal and robust memoryless discrimination from dependent observations discrimination is considered between two possible sources based on dependent observations of their output the discrimination problem is modeled by means of a general binary hypothesis test the main emphasis being on situations that cannot be modeled as signals in additive noise the observations are modeled as stationary m dependent or rho mixing processes the structure of the discriminator is such that the observations are passed through a memoryless nonlinearity summed up to form a test statistic which is then compared to a threshold only fixed sample size tests are considered four different performance measures which resemble the signal to noise ratios encountered in the signal in additive noise problems are derived under different problem formulations the optimal nonlinearities for each of the performance measures are derived as solutions to various integral equations for three of the four performance measures the authors have successfully obtained robust nonlinearities for uncertainty in the marginal and the joint probability density functions of the observations computer simulation results that demonstrate the advantage of using these nonlinearities over the i i d nonlinearity under the probability of error criterion are presented

linkage of modules over cohen macaulay rings abstract inspired by the theory of linkage for ideals the concept of sliding depth of a finitely generated module over a noetherian local ring is defined in terms of its ext modules as a result in the module theoretic linkage theory of martsinkovsky and strooker one proves the cohen macaulayness of a linked module if the base ring is cohen macaulay not necessarily gorenstein some interplay is established between the sliding depth condition and other module theoretic notions such as the g dimension and the property of being sequentially cohen macaulay

liaison classes of modules abstract we propose a concept of module liaison that extends gorenstein liaison of ideals and provides an equivalence relation among unmixed modules over a commutative gorenstein ring analyzing the resulting equivalence classes we show that several results known for gorenstein liaison are still true in the more general case of module liaison in particular we construct two maps from the set of even liaison classes of modules of fixed codimension into stable equivalence classes of certain reflexive modules as a consequence we show that the intermediate cohomology modules and properties like being perfect cohen macaulay buchsbaum or surjective buchsbaum are preserved in even module liaison classes furthermore we prove that the module liaison class of a complete intersection of codimension one consists of precisely all perfect modules of codimension one

statistical mechanics of learning multiple orthogonal signals asymptotic theory and fluctuation effects the learning of signal directions in high dimensional data through orthogonal decomposition or principal component analysis pca has many important applications in physics and engineering disciplines e g wireless communication information theory and econophysics the accuracy of the orthogonal decomposition can be studied using mean field theory previous analysis of data produced from a model with a single signal direction has predicted a retarded learning phase transition below which learning is not possible i e if the signal is too weak or the data set is too small then it is impossible to learn anything about the signal direction or magnitude in this contribution we show that the result can be generalized to the case where there are multiple signal directions each nondegenerate signal is associated with a retarded learning transition however fluctuations around the mean field solution lead to large finite size effects unless the signal strengths are very well separated we evaluate the one loop contribution to the mean field theory which shows that signal directions are indistinguishable from one another if their corresponding population eigenvalues are separated by o n tau with exponent tau 1 3 where n is the data dimension numerical simulations are consistent with the analysis and show that finite size effects can persist even for very large data sets

a maximum variance model for robust detection and estimation with dependent data a model that accounts for uncertain data dependency is developed by generating a large class of stationary stochastic processes each with the same univariate distribution this class can be considered to be a contamination class about the nominal independent and identically distributed i i d process distribution the class is developed specifically for application to robust detector and estimator design based on asymptotic variance application of this dependency class leads to an intuitively pleasing result the minimax variance estimators and the maximin efficacy detectors are the same as obtained using i i d asymptotic estimation and detection theory thus our technique generalizes previously obtained robust design results for i i d data to this dependent data case

statistical dynamics of on line independent component analysis the learning dynamics of on line independent component analysis is analysed in the limit of large data dimension we study a simple hebbian learning algorithm that can be used to separate out a small number of non gaussian components from a high dimensional data set the de mixing matrix parameters are confined to a stiefel manifold of tall orthogonal matrices and we introduce a natural gradient variant of the algorithm which is appropriate to learning on this manifold for large input dimension the parameter trajectory of both algorithms passes through a sequence of unstable fixed points each described by a diffusion process in a polynomial potential choosing the learning rate too large increases the escape time from each of these fixed points effectively trapping the learning in a sub optimal state in order to avoid these trapping states a very low learning rate must be chosen during the learning transient resulting in learning time scales of o n2 or o n3 iterations where n is the data dimension escape from each sub optimal state results in a sequence of symmetry breaking events as the algorithm learns each source in turn this is in marked contrast to the learning dynamics displayed by related on line learning algorithms for multilayer neural networks and principal component analysis although the natural gradient variant of the algorithm has nice asymptotic convergence properties it has an equivalent transient dynamics to the standard hebbian algorithm

linkage and the koszul homology of ideals 0 introduction let r be a local ring and i an ideal of r associated to i are several graded algebras the symmetric algebra of the module i sym i the rees algebra of i defined to be the algebra r i gn o inf and the associated graded algebra of i gr1 r rii g ii 2 in in 1 i i i our purpose in this paper is to study the behavior of these graded algebras when we change i through linkage recall the definition of linkage found in the paper of peskine and szpiro 22

one step memory nonlinearities for signal detection and discrimination from correlated observations detectors based on one step memory nonlinearities are introduced problems of discrimination between two arbitrary stationary m dependent or mixing sequences of observations and problems of detecting a weak signal in additive stationary m dependent or mixing noise are considered in this context for each problem the nonlinearity is optimized for performance criteria such as the generalized signal to noise ratio snr and the efficacy and is obtained as the solution to an appropriate linear integral equation the schemes considered can be made robust to statistical uncertainties determined by two alternating capacity classes for the second order joint pdfs probability distribution functions of the observations and by bounds on the correlation coefficients of time shifts of the observation sequence for the third and fourth order joint pdfs evaluation of the performance of the schemes via simulation reveals significant gains over that of detectors employing memoryless nonlinearities or the independent and identically distributed nonlinearity

even linkage classes in this paper we generalize the s and af type resolutions used by martin deschamps and perrin for curves in ip3 to subschemes of pure codimension in projective space and shows that these resolutions are interchanged by the mapping cone procedure under a simple linkage via these resolutions rao s correspondence is extended to give a bijection between even linkage classes of subschemes of pure codimension two and stable equivalence classes of reflexive sheaves s satisfying hi 5 0 and ext1 ev 9 0 further these resolutions are used to extend the work of martin deschamps and perrin for cohen macaulay curves in 1p3 to subschemes of pure codimension two in ip in particular even linkage classes of such subschemes satisfy the lazarsfeld rao property and any minimal subscheme for an even linkage class links directly to a minimal subscheme for the dual class

a nonlocal approach for asymptotic memoryless detection theory corresp previous literature on detector design has presented many tractable results using efficacy as a performance functional which is a small signal or local approach some nonlocal issues of memoryless detection are considered we argue that the asymptotic exponential rate of decrease of error probabilities or bayes risk provides the appropriate asymptotic measure of performance a main contribution is the development of an approximation for nonlocal design that retains the tractability of the efficacy approach we have proved the local consistency of our approximation also as a consequence of the analysis new proofs for pitman noether like theorems have been obtained

licci squarefree monomial ideals generated in degree two or with deviation two abstract we study squarefree monomial ideals which are in the linkage class of a complete intersection we call such an ideal licci firstly we classify all licci edge ideals secondly we prove that any cohen macaulay almost complete intersection squarefree monomial ideal is licci thirdly we characterize licci squarefree monomial ideals of deviation 2 in terms of hypergraphs associated to the ideals as an application of the classification in kimura terai and yoshida 2009 15

learning by on line gradient descent we study on line gradient descent learning in multilayer networks analytically and numerically the training is based on randomly drawn inputs and their corresponding outputs as defined by a target rule in the thermodynamic limit we derive deterministic differential equations for the order parameters of the problem which allow an exact calculation of the evolution of the generalization error first we consider a single layer perceptron with sigmoidal activation function learning a target rule defined by a network of the same architecture for this model the generalization error decays exponentially with the number of training examples if the learning rate is sufficiently small however if the learning rate is increased above a critical value perfect learning is no longer possible for architectures with hidden layers and fixed hidden to output weights such as the parity and the committee machine we find additional effects related to the existence of symmetries in these problems

the dynamics of on line principal component analysis the learning dynamics of an on line algorithm for principal component analysis is described exactly in the thermodynamic limit by means of coupled ordinary differential equations for a set of order parameters it is demonstrated that learning is delayed significantly because existing symmetries among student vectors have to be broken a closely related effect is the perfect or partial loss of initial knowledge in the course of learning the analysis shows that different learning rates for the student vectors improve the performance of the algorithm drastically

self averaging and on line learning conditions are given under which one may prove that the stochastic dynamics of on line learning can be described by the deterministic evolution of a finite set of order parameters in the thermodynamic limit a global constraint on the average magnitude of the increments in the stochastic process is necessary to ensure self averaging in the absence of such a constraint convergence may only be in probability

on stochastic approximation of the eigenvectors and eigenvalues of the expectation of a random matrix abstract in applications of signal processing and pattern recognition eigenvectors and eigenvalues of the statistical mean of a random matrix sequence are needed iterative methods are suggested and analyzed in which no sample moments are used convergence is shown by stochastic approximation theory

optimal generalization in perceptions a new learning algorithm for the one layer perceptron is presented it aims to maximize the generalization gain per example analytical results are obtained for the case of single presentation of each example the weight attached to a hebbian term is a function of the expected stability of the example in the teacher perceptron this leads to the obtention of upper bounds for the generalization ability this scheme can be iterated and the results of numerical simulations show that it converges within errors to the theoretical optimal generalization ability of the bayes algorithm analytical and numerical results for an algorithm with maximized generalization in the learning strategy with selection of examples are obtained and it is proved that as expected orthogonal selection is optimal exponential decay of the generalization error is obtained for the single presentation of selected examples

statistical mechanics of unsupervised structure recognition a model of unsupervised learning is studied where the environment provides n dimensional input examples that are drawn from two overlapping gaussian clouds we consider the optimization of two different objective functions the search for the direction of the largest variance in the data and the largest separating gap stability between clusters of examples respectively by means of a statistical mechanics analysis we investigate how well the underlying structure is inferred from a set of examples the performances of the learning algorithms depend crucially on the actual shape of the input distribution a generic result is the existence of a critical number of examples needed for successful learning the learning strategies are compared with methods different in spirit such as the estimation of parameters in a model distribution and an information theoretical approach

stochastic trapping in a solvable model of on line independent component analysis previous analytical studies of on line independent component analysis ica learning rules have focused on asymptotic stability and efficiency in practice the transient stages of learning are often more significant in determining the success of an algorithm this is demonstrated here with an analysis of a hebbian ica algorithm which can find a small number of nongaussian components given data composed of a linear mixture of independent source signals an idealized data model is considered in which the sources comprise a number of nongaussian and gaussian sources and a solution to the dynamics is obtained in the limit where the number of gaussian sources is infinite previous stability results are confirmed by expanding around optimal fixed points where a closed form solution to the learning dynamics is obtained however stochastic effects are shown to stabilize otherwise unstable suboptimal fixed points conditions required to destabilize one such fixed point are obtained for the case of a single nongaussian component indicating that the initial learning rate required to escape successfully is very low o n 2 where n is the data dimension resulting in very slow learning typically requiring o n3 iterations simulations confirm that this picture holds for a finite system

linkage of finite gorenstein dimension modules abstract for a horizontally linked module over a commutative semiperfect noetherian ring r the connections of its invariants reduced grade gorenstein dimension and depth are studied it is shown that under certain conditions the depth of a horizontally linked module is equal to the reduced grade of its linked module the connection of the serre condition s n on an r module of finite gorenstein dimension with the vanishing of the local cohomology groups of its linked module is discussed