super lie groups global topology and local structure a general mathematical framework for the super lie groups of supersymmetric theories is presented the definition of super lie group is given in terms of supermanifolds and two theorems analogous to theorems in classical lie group theory are proved the relationship of the super lie groups defined here to the formal groups of berezin and kac and the graded lie groups of kostant is analyzed

global convergence of oja s pca learning algorithm with a non zero approaching adaptive learning rate a non zero approaching adaptive learning rate is proposed to guarantee the global convergence of oja s principal component analysis pca learning algorithm most of the existing adaptive learning rates for oja s pca learning algorithm are required to approach zero as the learning step increases however this is not practical in many applications due to the computational round off limitations and tracking requirements the proposed adaptive learning rate overcomes this shortcoming the learning rate converges to a positive constant thus it increases the evolution rate as the learning step increases this is different from learning rates which approach zero which slow the convergence considerably and increasingly with time rigorous mathematical proofs for global convergence of oja s algorithm with the proposed learning rate are given in detail via studying the convergence of an equivalent deterministic discrete time ddt system extensive simulations are carried out to illustrate and verify the theory derived simulation results show that this adaptive learning rate is more suitable for oja s pca algorithm to be used in an online learning situation

energy function for the one unit oja algorithm the one unit oja algorithm plays a very important role in the study of principal component analysis neural networks in this paper we propose an energy function whose steepest descent direction i e negative gradient direction is the same as the average evolution direction of the one unit oja algorithm and the energy function has two global minimal points corresponding to the two converged points of the one unit oja algorithm and it has no other local minimal points

chern simons forms on principal superfiber bundles a graded weil homomorphism is defined for principal superfiber bundles and the related transgression or chern simons forms are introduced as an example of the application of these concepts a superextension of the dirac monopole is discussed

gibbsian dynamics and ergodicity for the stochastically forced navier stokes equation we study stationary measures for the two dimensional navier stokes equation with periodic boundary condition and random forcing we prove uniqueness of the stationary measure under the condition that all determining modes are forced the main idea behind the proof is to study the gibbsian dynamics of the low modes obtained by representing the high modes as functionals of the time history of the low modes

the axioms of supermanifolds and a new structure arising from them on presente une analyse des supervarietes sur un algebre arbitraire commutative graduee en partant d un ensemble d axiomes dont le premier est que les derivations de la structure faisceau d une supervariete sont localement libres

cohomology of the structure sheaf of real and complex supermanifolds the cohomology of the structure sheaf of real and complex supermanifolds is studied it is found to be nontrivial also in the real case unless the supermanifold is de witt i e it is a fiber bundle on an ordinary manifold with a vector fiber as a consequence the dolbeault theorem can be extended only to complex de witt supermanifolds the relevance of the nontrivial cohomology of the structure sheaf to the classification of complex line superbundles is discussed the relationship between the picard group of a complex de witt supermanifold and the picard group of its body is shown

convergence analysis of a deterministic discrete time system of oja s pca learning algorithm the convergence of oja s principal component analysis pca learning algorithms is a difficult topic for direct study and analysis traditionally the convergence of these algorithms is indirectly analyzed via certain deterministic continuous time dct systems such a method will require the learning rate to converge to zero which is not a reasonable requirement to impose in many practical applications recently deterministic discrete time ddt systems have been proposed instead to indirectly interpret the dynamics of the learning algorithms unlike dct systems ddt systems allow learning rates to be constant which can be a nonzero this paper will provide some important results relating to the convergence of a ddt system of oja s pca learning algorithm it has the following contributions 1 a number of invariant sets are obtained based on which we can show that any trajectory starting from a point in the invariant set will remain in the set forever thus the nondivergence of the trajectories is guaranteed 2 the convergence of the ddt system is analyzed rigorously it is proven in the paper that almost all trajectories of the system starting from points in an invariant set will converge exponentially to the unit eigenvector associated with the largest eigenvalue of the correlation matrix in addition exponential convergence rate are obtained providing useful guidelines for the selection of fast convergence learning rate 3 since the trajectories may diverge the careful choice of initial vectors is an important issue this paper suggests to use the domain of unit hyper sphere as initial vectors to guarantee convergence 4 simulation results will be furnished to illustrate the theoretical results achieved

cohomology of supermanifolds the aim of this chapter is to unfold a basic cohomological theory for supermanifolds which will be exploited in the next chapter to study the structure of superbundles in particular to build a theory of characteristic classes this cohomology theory does not embody only trivial extensions of results valid for differentiable manifolds for instance the natural analogue of the de rham theorem does not hold in general and similarly in the case of complex supermanifolds there is generally speaking no analogue of the dolbeault theorem these features are consequences of the fact that the structure sheaf of a supermanifold does not need to be cohomologically trivial related to this is also the fact that the cohomology of the complex of global graded differential forms on a g supermanifold m a i e the super de rham cohomology of m a depends on the g supermanifold structure of m a so that homeomorphic and even smoothly diffeomorphic g supermanifolds may have a different super de rham cohomology that is super de rham cohomology is a fine invariant of the supermanifold structure

characteristic classes of super vector bundles a theory of characteristic classes for complex super vector bundles over supermanifolds is developed such characteristic classes are proved to fulfill the usual properties and it is shown that under suitable conditions they can be represented in terms of supersmooth curvature forms

exponential convergence for the stochastically forced navier stokes equations and other partially dissipative dynamics we prove that the two dimensional navier stokes equations possess an exponentially attracting invariant measure this result is in fact the consequence of a more general harris like ergodic theorem applicable to many dissipative stochastic pdes and stochastic processes with memory a simple iterated map example is also presented to help build intuition and showcase the central ideas in a less encumbered setting to analyze the iterated map a general doeblin like theorem is proven one of the main features of this paper is the novel coupling construction used to examine the ergodic theory of the non markovian processes

adaptive minor component extraction with modular structure an information criterion for adaptively estimating multiple minor eigencomponents of a covariance matrix is proposed it is proved that the proposed criterion has a unique global minimum at the minor subspace and that all other equilibrium points are saddle points based on the gradient search approach of the proposed information criterion an adaptive algorithm called adaptive minor component extraction amex is developed the proposed algorithm automatically performs the multiple minor component extraction in parallel without the inflation procedure similar to the adaptive lattice filter structure the amex algorithm also has the flexibility wherein increasing the number of the desired minor component does not affect the previously extracted minor components the amex algorithm has a highly modular structure and the various modules operate completely in parallel without any delay simulation results are given to demonstrate the effectiveness of the amex algorithm for both the minor component analysis mca and the minor subspace analysis msa

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

ergodicity of the 2 d navier stokes equation under random perturbations a 2 dimensional navier stokes equation perturbed by a sufficiently distributed white noise is considered existence of invariant measures is known from previous works the aim is to prove uniqueness of the invariant measures strong law of large numbers and convergence to equilibrium

stochastic equations in hilbert space with application to navier stokes equations in any dimension abstract we give an existence theorem for an abstract nonlinear stochastic evolution equation in a hilbert space the result is applicable to the stochastic navier stokes equation in any dimension with a nonlinear noise term

graded manifolds supermanifolds and infinite dimensional grassmann algebras a new approach to superdifferentiable functions of grassmann variables is developed which avoids ambiguities in odd derivatives this is used to give an improved definition of supermanifold over a finite dimensional grassmann algebra a natural embedding of super manifolds over grassmann algebras with increasing number l of generators is developed and thus a limit asl tends to infinity is possible a correspondence between graded manifolds and supermanifolds is constructed extending results of 5 and 8

exponential mixing of the 2d stochastic navier stokes dynamics we consider the navier stokes equation on a two dimensional torus with a random force which is white noise in time and excites only a finite number of modes the number of excited modes depends on the viscosity v and grows like v 3 when v goes to zero we prove that this markov process has a unique invariant measure and is exponentially mixing in time

martingale and stationary solutions for stochastic navier stokes equations we prove the existence of martingale solutions and of stationary solutions of stochastic navier stokes equations under very general hypotheses on the diffusion term the stationary martingale solutions yield the existence of invariant measures when the transition semigroup is well defined the results are obtained by a new method of compactness

exponential ergodicity for stochastic burgers and 2d navier stokes equations abstract it is shown that transition measures of the stochastic navier stokes equation in 2 d converge exponentially fast to the corresponding invariant measures in the distance of total variation as a corollary we obtain the existence of spectral gap for a related semigroup obtained by a sort of ground state transformation analogous results are proved for the stochastic burgers equation