ranks of elliptic curves this paper gives a general survey of ranks of elliptic curves over the field of rational numbers the rank is a measure of the size of the set of rational points the paper includes discussions of the birch and swinnertondyer conjecture the parity conjecture ranks in families of quadratic twists and ways to search for elliptic curves of large rank

organizing the arithmetic of elliptic curves abstract suppose that e is an elliptic curve defined over a number field k p is a rational prime and k is the maximal z p power extension of k in previous work b mazur k rubin elliptic curves and class field theory in ta tsien li ed proceedings of the international congress of mathematicians icm 2002 vol ii higher education press beijing 2002 pp 185 195 b mazur k rubin pairings in the arithmetic of elliptic curves in j cremona et al eds modular curves and abelian varieties progress in mathematics vol 224 2004 pp 151 163 we discussed the possibility that much of the arithmetic of e over k i e the mordell weil groups and their p adic height pairings the shafarevich tate groups and their cassels pairings over all finite extensions of k in k can be described efficiently in terms of a single skew hermitian matrix with entries drawn from the iwasawa algebra of k k in this paper using work of nekovar j nekovar selmer complexes preprint available at http www math jussieu fr nekovar pu we show that under not too stringent conditions such an organizing matrix does in fact exist we also work out an assortment of numerical instances in which we can describe the organizing matrix explicitly

p converse to a theorem of gross zagier kolyvagin and rubin let e be a cm elliptic curve over the rationals and p 3 a good ordinary prime for e we show that begin aligned mathrm corank mathbb z p mathrm sel p infty e mathbb q 1 implies mathrm ord s 1 l s e mathbb q 1 end aligned for the p infty selmer group mathrm sel p infty e mathbb q and the complex l function l s e mathbb q in particular the tate shafarevich group hbox x e mathbb q is finite whenever mathrm corank mathbb z p mathrm sel p infty e mathbb q 1 we also prove an analogous p converse for cm abelian varieties arising from weight two elliptic cm modular forms with trivial central character for non cm elliptic curves over the rationals first general results towards such a p converse theorem are independently due to skinner a converse to a theorem of gross zagier and kolyvagin arxiv 1405 7294 2014 and zhang camb j math 2 2 191 253 2014

refinement equations and distributional fixed points abstract connections between l 1 solutions of the refinement equation of the form f x l f lx m dp and distributional fixed points of a special random affine map are investigated using results on convergence of perpetuities the direct form of the fourier transform of l 1 solutions of the above refinement equation is obtained

refinement type equations and grincevicjus series abstract we consider l 1 solutions of the following refinement type equations f x n z c n 1 f k x n n z c n 1 f k x n where k 2 is an integer and for all n z reals c n 1 c n 1 are non negative with n z c n 1 c n 1 k and n z log n c n 1 c n 1 necessary and sufficient conditions for the existence of non trivial l 1 solutions in several special cases are given

on l1 solutions of a two direction refinement equation abstract in this paper we examine l 1 solutions of the following two direction refinement equation f x n n n c n 1 f k x n n n n c n 1 f k x n we prove that the vector space of all l 1 solutions of the above equation is at most one dimensional and consists of compactly supported functions of constant sign we also show that in many interesting cases any l 1 solution of the two direction refinement equation is either positive or negative on its support next we present sufficient conditions easy for verification for the existence of nontrivial l 1 solutions of the two direction refinement equation as well as for the nonexistence of such solutions

optimal approximation schedules for a class of iterative algorithms with an application to multigrid value iteration many iterative algorithms employ operators which are difficult to evaluate exactly but for which a graduated range of approximations exist in such cases coarse to fine algorithms are often used in which a crude initial operator approximation is gradually refined with new iterations in such algorithms because the computational complexity increases over iterations the algorithm s convergence rate is properly calculated with respect to cumulative computation time this suggests the problem of determining an optimal rate of refinement for the operator approximation this paper shows that for linearly convergent algorithm the optimal rate of refinement approaches the rate of convergence of the exact algorithm itself regardless of the tolerance complexity relationship we illustrate this result with an analysis of coarse to fine grid algorithms for markov decision processes with continuous state spaces using the methods proposed here we deduce an algorithm that presents optimal complexity results and consists solely of a non adaptive schedule for the gradual decrease of grid size

a study on orthogonal two direction vector valued wavelets and two direction wavelet packets abstract in this paper the notion of two direction vector valued multiresolution analysis and the two direction orthogonal vector valued wavelets are introduced the definition for two direction orthogonal vector valued wavelet packets is proposed an algorithm for constructing a class of two direction orthogonal vector valued compactly supported wavelets corresponding to the two direction orthogonal vector valued compactly supported scaling functions is proposed by virtue of matrix theory and time frequency analysis method the properties of the two direction vector valued wavelet packets are investigated at last the direct decomposition relation for space l 2 r r is presented

iwasawa theory of elliptic curves at supersingular primes over ℤ p extensions of number fields in this paper we make a study of the iwasawa theory of an elliptic curve at a supersingular prime p along an arbitrary zp extension of a number field k in the case when p splits completely in k generalizing work of kobayashi 9 and perrin riou 17 we define restricted selmer groups and invariants we then derive asymptotic formulas describing the growth of the selmer group in terms of these invariants to be able to work with non cyclotomic zp extensions a new local result is proven that gives a complete description of the formal group of an elliptic curve at a supersingular prime along any ramified zp extension of qp

nearly optimal control laws for nonlinear systems with saturating actuators using a neural network hjb approach the hamilton jacobi bellman hjb equation corresponding to constrained control is formulated using a suitable nonquadratic functional it is shown that the constrained optimal control law has the largest region of asymptotic stability ras the value function of this hjb equation is solved for by solving for a sequence of cost functions satisfying a sequence of lyapunov equations le a neural network is used to approximate the cost function associated with each le using the method of least squares on a well defined region of attraction of an initial stabilizing controller as the order of the neural network is increased the least squares solution of the hjb equation converges uniformly to the exact solution of the inherently nonlinear hjb equation associated with the saturating control inputs the result is a nearly optimal constrained state feedback controller that has been tuned a priori off line

derived p adic heights and p adic l functions if e is an elliptic curve defined over a number field and p is a prime of good ordinary reduction for e a theorem of rubin relates the p adic height pairing on the p power selmer group of e to the first derivative of a cohomologically defined p adic l function attached to e bertolini and darmon have defined a sequence of derived p adic heights in this paper we give an alternative definition of the p adic height pairing and prove a generalization of rubin s result relating the derived heights to higher derivatives of p adic l functions we also relate degeneracies in the derived heights to the failure of the selmer group of e over a z p extension to be semi simple as an iwasawa module generalizing results of perrin riou

adaptive dynamic programming for finite horizon optimal control of discrete time nonlinear systems with varepsilon error bound in this paper we study the finite horizon optimal control problem for discrete time nonlinear systems using the adaptive dynamic programming adp approach the idea is to use an iterative adp algorithm to obtain the optimal control law which makes the performance index function close to the greatest lower bound of all performance indices within an error bound the optimal number of control steps can also be obtained by the proposed adp algorithms a convergence analysis of the proposed adp algorithms in terms of performance index function and control policy is made in order to facilitate the implementation of the iterative adp algorithms neural networks are used for approximating the performance index function computing the optimal control policy and modeling the nonlinear system finally two simulation examples are employed to illustrate the applicability of the proposed method

proto value functions a laplacian framework for learning representation and control in markov decision processes this paper introduces a novel spectral framework for solving markov decision processes mdps by jointly learning representations and optimal policies the major components of the framework described in this paper include i a general scheme for constructing representations or basis functions by diagonalizing symmetric diffusion operators ii a specific instantiation of this approach where global basis functions called proto value functions pvfs are formed using the eigenvectors of the graph laplacian on an undirected graph formed from state transitions induced by the mdp iii a three phased procedure called representation policy iteration comprising of a sample collection phase a representation learning phase that constructs basis functions from samples and a final parameter estimation phase that determines an approximately optimal policy within the linear subspace spanned by the current basis functions iv a specific instantiation of the rpi framework using least squares policy iteration lspi as the parameter estimation method v several strategies for scaling the proposed approach to large discrete and continuous state spaces including the nystrom extension for out of sample interpolation of eigenfunctions and the use of kronecker sum factorization to construct compact eigenfunctions in product spaces such as factored mdps vi finally a series of illustrative discrete and continuous control tasks which both illustrate the concepts and provide a benchmark for evaluating the proposed approach many challenges remain to be addressed in scaling the proposed framework to large mdps and several elaboration of the proposed framework are briefly summarized at the end

variation of heegner points in hida families given a weight two modular form f with associated p adic galois representation v f for certain quadratic imaginary fields k one can construct canonical classes in the galois cohomology of v f by taking the kummer images of heegner points on the modular abelian variety attached to f we show that these classes can be interpolated as f varies in a hida family and construct an euler system of big heegner points for hida s universal ordinary deformation of v f we show that the specialization of this big euler system to any form in the hida family is nontrivial extending results of cornut and vatsal from modular forms of weight two and trivial character to all ordinary modular forms and propose a horizontal nonvanishing conjecture for these cohomology classes the horizontal nonvanishing conjecture implies via the theory of euler systems a conjecture of greenberg on the generic ranks of selmer groups in hida families

data driven approximate value iteration with optimality error bound analysis abstract features of the data driven approximate value iteration avi algorithm proposed in li et al 2014 for dealing with the optimal stabilization problem include that only process data is required and that the estimate of the domain of attraction for the closed loop is enlarged however the controller generated by the data driven avi algorithm is an approximate solution for the optimal control problem in this work a quantitative analysis result on the error bound between the optimal cost and the cost under the designed controller is given this error bound is determined by the approximation error of the estimation for the optimal cost and the approximation error of the controller function estimator the first one is concretely determined by the approximation error of the data driven dynamic programming dp operator to the dp operator and the approximation error of the value function estimator these three approximation errors are zeros when the data set of the plant is sufficient and infinitely complete and the number of samples in the interested state space is infinite this means that the cost under the designed controller equals to the optimal cost when the number of iterations is infinite

transforms associated to square integrable group representations i general results let g be a locally compact group which need not be unimodular let x u x x g be an irreducible unitary representation of g in a hilbert space h u assume that u is square integrable i e that there exists in h u at least one nonzero vector g such that u x g g 2 dx we give here a reasonably self contained analysis of the correspondence associating to every vector f h u the function u x g f on g discussing its isometry characterization of the range inversion and simplest interpolation properties this correspondence underlies many properties of generalized coherent states

finding large selmer groups in this paper we show how to use a recent theorem of nekova r 12 to produce families of examples of elliptic curves over number fields whose p power selmer groups grow systematically in z d extensions we give a somewhat different exposition and proof of nekova r s theorem and we show in many cases how to replace the fundamental requirement that the elliptic curve has odd p selmer rank by a root number calculation

subvarieties of shimura varieties 7 proof of the main result 19 7 3 the case where is bounded 20 7 4 the case where is not bounded 22 both authors were partially supported by the european research training network contract hprn ct 200000120 arithmetic algebraic geometry adresses irmar campus de beaulieu 35042 rennes cedex france edix maths univ rennes1 fr ayafaev maths univ rennes1 fr

construction of biorthogonal two direction refinable function and two direction wavelet with dilation factor m an algorithm for constructing a pair of biorthogonal two direction refinable function and the corresponding biorthogonal two direction wavelet is presented in particular the algorithm can also be used to construct two direction orthogonal refinable function finally we give an example illustrating how to use our method to construct biorthogonal two direction scaling function with dilation factor 3

on compactly supported spline wavelets and a duality principle let v 1 v 0 v 1 be a multiresolution analysis of l 2 generated by the mth order b spline n m x we exhibit a compactly supported basic wavelet m x that generates the corresponding orthogonal complementary wavelet subspaces w 1 w 0 w 1 consequently the two finite sequences that describe the two scale relations of n m x and m x in terms of n m 2x j j z yield an efficient reconstruction algorithm to give an efficient wavelet decomposition algorithm based on these two finite sequences we derive a duality principle which also happens to yield the dual bases n m x j and 59 m x j relative to n m x j and m x j respectively

p adic heights of heegner points and λ adic regulators let e be an elliptic curve defined over q the aim of this paper is to make it possible to compute heegner l functions and anticyclotomic adic regulators of e which were studied by mazur rubin and howard we generalize results of cohen and watkins and thereby compute heegner points of nonfundamental discriminant we then prove a relationship between the denominator of a point of e defined over a number field and the leading coefficient of the minimal polynomial of its xcoordinate using this relationship we recast earlier work of mazur stein and tate to produce effective algorithms to compute p adic heights of points of e defined over number fields these methods enable us to give the first explicit examples of heegner l functions and anticyclotomic adic regulators

dilation equations and markov operators abstract using the theory of markov operators we give a new proof of the known fact saying that for every positive integers n and k 1 and for every nonnegative reals c 0 c n satisfying the first sum rule the dilation equation f x n 0 n c n f k x n has a nontrivial l 1 solution

approximate robust policy iteration using multilayer perceptron neural networks for discounted infinite horizon markov decision processes with uncertain correlated transition matrices we study finite state finite action discounted infinite horizon markov decision processes with uncertain correlated transition matrices in deterministic policy spaces existing robust dynamic programming methods cannot be extended to solving this class of general problems in this paper based on a robust optimality criterion an approximate robust policy iteration using a multilayer perceptron neural network is proposed it is proven that the proposed algorithm converges in finite iterations and it converges to a stationary optimal or near optimal policy in a probability sense in addition we point out that sometimes even a direct enumeration may not be applicable to addressing this class of problems however a direct enumeration based on our proposed maximum value approximation over the parameter space is a feasible approach we provide further analysis to show that our proposed algorithm is more efficient than such an enumeration method for various scenarios

special values of anticyclotomic l functions the object of this paper is to extend the results and methods of vat01 where it was shown how cases of a conjecture of mazur on the behavior of l functions in an anticyclotomic zp extension could be deduced by studying the distribution of heegner points associated to definite quaternion algebras the main result of vat01 showed that when the sign in the functional equation is 1 then the special values of the l functions in question are generically nonzero in this paper we propose to study the special values modulo a prime of q and can offer three new theorems in this direction the first result theorem 1 1 determines the iwasawa invariant of the p adic l functions of bertolini and darmon we show that is usually zero but not always when 0 we give a precise formula for the value and an interpretation of the positivity in terms of congruences the second result theorem 1 2 pertains to the case where has residue characteristic p in this case we show that the l values are typically units but again that this is not always the case for technical reasons this theorem applies only to a restricted class of but it seems rather likely that the restrictions can be lifted finally we use the result of theorem 1 2 to transfer our results from the case where the sign in the functional equation is 1 to the case of sign 1 and to derivatives of l functions this is achieved by using congruences and the sign change phenomenon exploited by bertolini darmon namely we prove a jochnowitz congruence which relates the nontriviality of a special value modulo to the nontriviality of a classical heegner point on a modular curve the result is stated in theorem 1 4 to describe these results more precisely let g denote a newform on 0 n let k denote an imaginary quadratic field of discriminant d such that d is prime to n write n n n where n is divisible only by those primes which are split in k whereas n is divisible only by primes

matrix refinement type equations abstract taking advantage of perpetuities and the asymptotic behavior of products of random matrices we obtain the direct form of the fourier transform of an l 1 solution of the following random matrix refinement type equation f x det l c f l x m p d where m r p is a random vector and l r p p c r q q are random matrices moreover the dimension of the space of all l 1 solutions of this refinement type equation is examined

neural network based near optimal control for a class of discrete time affine nonlinear systems with control constraints in this paper the near optimal control problem for a class of nonlinear discrete time systems with control constraints is solved by iterative adaptive dynamic programming algorithm first a novel nonquadratic performance functional is introduced to overcome the control constraints and then an iterative adaptive dynamic programming algorithm is developed to solve the optimal feedback control problem of the original constrained system with convergence analysis in the present control scheme there are three neural networks used as parametric structures for facilitating the implementation of the iterative algorithm two examples are given to demonstrate the convergence and feasibility of the proposed optimal control scheme

h _ infty tracking control of completely unknown continuous time systems via off policy reinforcement learning this paper deals with the design of an h infty tracking controller for nonlinear continuous time systems with completely unknown dynamics a general bounded l 2 gain tracking problem with a discounted performance function is introduced for the h infty tracking a tracking hamilton jacobi isaac hji equation is then developed that gives a nash equilibrium solution to the associated min max optimization problem a rigorous analysis of bounded l 2 gain and stability of the control solution obtained by solving the tracking hji equation is provided an upper bound is found for the discount factor to assure local asymptotic stability of the tracking error dynamics an off policy reinforcement learning algorithm is used to learn the solution to the tracking hji equation online without requiring any knowledge of the system dynamics convergence of the proposed algorithm to the solution to the tracking hji equation is shown simulation examples are provided to verify the effectiveness of the proposed method

reinforcement learning with automatic basis construction based on isometric feature mapping value function approximation vfa has been a major research topic in reinforcement learning although various reinforcement learning algorithms with vfa have been proposed the performance of most previous algorithms depends on the predefined structure of the basis functions to address this problem this paper presents a novel basis learning method for vfa based on isometric feature mapping ifm in the proposed method basis functions for vfa are automatically generated by constructing the optimal embedding basis of the data in a d dimensional euclidean space which best preserves the estimated intrinsic geometry of the manifold furthermore the ifm based basis learning method is integrated with approximation policy iteration api for learning control in markov decision problems with large state spaces a new manifold reinforcement learning framework termed ifm based api ifm api is presented three learning control problems including a real control system of the googol single inverted pendulum were studied to evaluate the performance of the proposed ifm api algorithm the simulation and experimental results show that compared with other basis selection or learning methods the ifm based basis learning method can automatically compute an efficient set of basis functions with much fewer predefined parameters and less computational costs besides it is illustrated that the proposed ifm api algorithm can obtain better learning control policies than other api methods

uniform refinement of curves abstract we propose and analyze a class of algorithms for the generation of curves and surfaces these algorithms encompass some well known methods of subdivision for bernstein bezier curves de casteljau s algorithm and b spline curves lane and riesenfeld s algorithm several results concerning properties of the limiting curves as well as related questions are discussed

equidistribution of heegner points and ternary quadratic forms we prove new equidistribution results for galois orbits of heegner points with respect to single reduction maps at inert primes the arguments are based on two different techniques primitive representations of integers by quadratic forms and distribution relations for heegner points our results generalize an equidistribution result with respect to a single reduction map established by cornut and vatsal in the sense that we allow both the fundamental discriminant and the conductor to grow moreover for fixed fundamental discriminant and variable conductor we deduce an effective surjectivity theorem for the reduction map from heegner points to supersingular points at a fixed inert prime our results are applicable to the setting considered by kolyvagin in the construction of the heegner points euler system

two direction refinable functions and two direction wavelets with dilation factor m abstract motivated by cardinal b spline scaling functions we introduce the concept of two direction refinable functions and two direction wavelets with dilation factor m we investigate the following two direction refinement equation x k p k mx k k p k k mx where m 2 is an integer here two sequences p k k and p k k are called the positive mask and negative mask respectively based on the positive mask and negative mask the conditions that the above equation has compactly distributional solutions are established the conditions that the above equation has l2 stable solutions are also presented the support of x is discussed amply an algorithm for constructing orthogonal two direction refinable functions and their two direction wavelets is presented based on orthogonal two direction scaling functions and two direction wavelets orthogonal and symmetric multiscaling functions and their multiwavelets can be constructed easily in the end a construction example is given

error bounds of adaptive dynamic programming algorithms for solving undiscounted optimal control problems in this paper we establish error bounds of adaptive dynamic programming algorithms for solving undiscounted infinite horizon optimal control problems of discrete time deterministic nonlinear systems we consider approximation errors in the update equations of both value function and control policy we utilize a new assumption instead of the contraction assumption in discounted optimal control problems we establish the error bounds for approximate value iteration based on a new error condition furthermore we also establish the error bounds for approximate policy iteration and approximate optimistic policy iteration algorithms it is shown that the iterative approximate value function can converge to a finite neighborhood of the optimal value function under some conditions to implement the developed algorithms critic and action neural networks are used to approximate the value function and control policy respectively finally a simulation example is given to demonstrate the effectiveness of the developed algorithms

construction of biorthogonal multiwavelets abstract there are perfect formulas for the constructions of biorthogonal uniwavelets let x k z p k 2x k x k z p k 2x k be a pair of biorthogonal uniscaling functions then a pair of biorthogonal uniwavelet associated with the above biorthogonal uniscaling functions can be easily expressed as x k z 1 k 1 p 1 k 2x k x k z 1 k 1 p 1 k 2x k however it seems that there is not such a good formula of similar structure for biorthogonal multiwavelets in this paper we will give a procedure for constructing compactly supported biorthogonal multiwavelets which makes construction of biorthogonal multiwavelets easy like in the construction of biorthogonal uniwavelet our approach is also suitable for the case of compactly supported orthogonal multiwavelets four examples for constructing multiwavelets are given

kernel based least squares policy iteration for reinforcement learning in this paper we present a kernel based least squares policy iteration klspi algorithm for reinforcement learning rl in large or continuous state spaces which can be used to realize adaptive feedback control of uncertain dynamic systems by using klspi near optimal control policies can be obtained without much a priori knowledge on dynamic models of control plants in klspi mercer kernels are used in the policy evaluation of a policy iteration process where a new kernel based least squares temporal difference algorithm called klstd q is proposed for efficient policy evaluation to keep the sparsity and improve the generalization ability of klstd q solutions a kernel sparsification procedure based on approximate linear dependency ald is performed compared to the previous works on approximate rl methods klspi makes two progresses to eliminate the main difficulties of existing results one is the better convergence and near optimality guarantee by using the klstd q algorithm for policy evaluation with high precision the other is the automatic feature selection using the ald based kernel sparsification therefore the klspi algorithm provides a general rl method with generalization performance and convergence guarantee for large scale markov decision problems mdps experimental results on a typical rl task for a stochastic chain problem demonstrate that klspi can consistently achieve better learning efficiency and policy quality than the previous least squares policy iteration lspi algorithm furthermore the klspi method was also evaluated on two nonlinear feedback control problems including a ship heading control problem and the swing up control of a double link underactuated pendulum called acrobot simulation results illustrate that the proposed method can optimize controller performance using little a priori information of uncertain dynamic systems it is also demonstrated that klspi can be applied to online learning control by incorporating an initial controller to ensure online performance

two direction multiwavelet moments abstract two direction multiscaling functions and two direction multiwavelets associated with are a more general and more flexible setting than one direction multiscaling functions and multiwavelets in this paper we derive two methods for computing continuous moments of orthogonal two direction multiscaling functions and multiwavelets the first method is by doubling and the second method is by separation examples for both methods are given

performance loss bounds for approximate value iteration with state aggregation we consider approximate value iteration with a parameterized approximator in which the state space is partitioned and the optimal cost to go function over each partition is approximated by a constant we establish performance loss bounds for policies derived from approximations associated with fixed points these bounds identify benefits to using invariant distributions of appropriate policies as projection weights such projection weighting relates to what is done by temporal difference learning our analysis also leads to the first performance loss bound for approximate value iteration with an average cost objective

relaxing dynamic programming the idea of dynamic programming is general and very simple but the curse of dimensionality is often prohibitive and restricts the fields of application this paper introduces a method to reduce the complexity by relaxing the demand for optimality the distance from optimality is kept within prespecified bounds and the size of the bounds determines the computational complexity several computational examples are considered the first is optimal switching between linear systems with application to design of a dc dc voltage converter the second is optimal control of a linear system with piecewise linear cost with application to stock order control finally the method is applied to a partially observable markov decision problem pomdp

finite horizon control constrained nonlinear optimal control using single network adaptive critics to synthesize fixed final time control constrained optimal controllers for discrete time nonlinear control affine systems a single neural network nn based controller called the finite horizon single network adaptive critic is developed in this paper inputs to the nn are the current system states and the time to go and the network outputs are the costates that are used to compute optimal feedback control control constraints are handled through a nonquadratic cost function convergence proofs of 1 the reinforcement learning based training method to the optimal solution 2 the training error and 3 the network weights are provided the resulting controller is shown to solve the associated time varying hamilton jacobi bellman equation and provide the fixed final time optimal solution performance of the new synthesis technique is demonstrated through different examples including an attitude control problem wherein a rigid spacecraft performs a finite time attitude maneuver subject to control bounds the new formulation has great potential for implementation since it consists of only one nn with single set of weights and it provides comprehensive feedback solutions online though it is trained offline

uniform distribution of heegner points let e be a modular elliptic curve over q of conductor n let k denote an imaginary quadratic field of discriminant d with n d 1 if p is a prime then there exists a unique zp extension k k such that gal k q acts nontrivially on gal k k the field k is called the anticyclotomic zp extension of k let e k denote the mordell weil group of e over k then a fundamental conjecture of mazur maz predicts that the size of e k is controlled by the prime factorization of n in k equivalently mazur s conjecture relates the size of the mordell weil group to the sign in the functional equation of certain l series the conjecture was verified by greenberg rohrlich and rubin in what mazur calls the exceptional case when e has complex multiplication by k more generally they settled the conjecture for certain abelian varieties with complex multiplication for a discussion of this cm case we refer the reader to gre roh and rub our goal in this paper is to treat the generic case which occurs either when e has no cm or when the field of complexmultiplications is distinct from k under certain conditions on e and k mazur s conjecture predicts that the group e k is finitely generated our main result asserts that this is in fact the case at least when p is an ordinary prime for e or when the class number of k is prime to p the main new ingredient we introduce is that of equidistribution following ideas used by ferrero andwashington to study the cyclotomic invariant more precisely we show that theheegner points associated to definite quaternion algebras are uniformly distributed on the components of a certain curve x and that the elements of a certain galois group act independently in a suitable sense this combined with a special value formula due to gross allows us to conclude that the special values of anticyclotomic l functions are almost always nonzero so that the statement about the mordell weil groups follows from the machinery of euler systems as developed by bertolini and darmon bd

biorthogonal two direction refinable function and two direction wavelet the concept of two direction refinable function and two direction wavelet is generalized to the biorthogonal setting the definition of biorthogonal two direction refinable function is introduced and the biorthogonality criteria for two direction refinable function is established we present an algorithm for constructing a pair of biorthogonal two direction refinable function and the corresponding biorthogonal two direction wavelet finally we give two examples illustrating how to use our method to construct a pair of biorthogonal refinable function and its corresponding biorthogonal two direction wavelet pair

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

a block spin construction of ondelettes part i lemarie functions using block spin assignments we construct anl2 orthonormal basis consisting of dyadic scalings and translates of just a finite number of functions these functions have exponential localization and for even values of a construction parameterm one can make them classcm 1 with vanishing moments up to orderm inclusive such a basis has an important application to phase cell cluster expansions in quantum field theory

neural network control based adaptive learning design for nonlinear systems with full state constraints in order to stabilize a class of uncertain nonlinear strict feedback systems with full state constraints an adaptive neural network control method is investigated in this paper the state constraints are frequently emerged in the real life plants and how to avoid the violation of state constraints is an important task by introducing a barrier lyapunov function blf to every step in a backstepping procedure a novel adaptive backstepping design is well developed to ensure that the full state constraints are not violated at the same time one remarkable feature is that the minimal learning parameters are employed in blf backstepping design by making use of lyapunov analysis we can prove that all the signals in the closed loop system are semiglobal uniformly ultimately bounded and the output is well driven to follow the desired output finally a simulation is given to verify the effectiveness of the method

performance bounds in l_p norm for approximate value iteration approximate value iteration avi is a method for solving large markov decision problems by approximating the optimal value function with a sequence of value function representations v n processed according to the iterations v n 1 mathcal at v n where mathcal t is the so called bellman operator and mathcal a an approximation operator which may be implemented by a supervised learning sl algorithm usual bounds on the asymptotic performance of avi are established in terms of the l infty norm approximation errors induced by the sl algorithm however most widely used sl algorithms such as least squares regression return a function the best fit that minimizes an empirical approximation error in l p norm p geq 1 in this paper we extend the performance bounds of avi to weighted l p norms which enables us to directly relate the performance of avi to the approximation power of the sl algorithm hence assuring the tightness and practical relevance of these bounds the main result is a performance bound of the resulting policies expressed in terms of the l p norm errors introduced by the successive approximations the new bound takes into account a concentration coefficient that estimates how much the discounted future state distributions starting from a probability measure used to assess the performance of avi can possibly differ from the distribution used in the regression operation we illustrate the tightness of the bounds on an optimal replacement problem

finding large selmer rank via an arithmetic theory of local constants we obtain lower bounds for selmer ranks of elliptic curves over dihedral extensions of number fields suppose k k is a quadratic extension of number fields e is an elliptic curve defined over k and p is an odd prime let denote the maximal abelian p extension of k that is unramified at all primes where e has bad reduction and that is galois over k with dihedral galois group i e the generator c of gal k k acts on gal k by inversion we prove under mild hypotheses on p that if the zp rank of the pro p selmer group sp e k is odd then rank zp s p e f f k for every finite extension f of k in

the subconvexity problem for rankin selberg l functions and equidistribution of heegner points ii we prove a general subconvex bound in the level aspect for rankin selberg l functions associated with two primitive holomorphic or maass cusp forms over q we use this bound to establish the equidistribution of incomplete galois orbits of heegner points on shimura curves associated with indefinite quaternion algebras over q

construction of nonseparable dual ω wavelet frames in l2 rs suppose j 1 a 1 j 1 a 1 and j 1 2 a 2 1 j 1 2 a 2 1 are two pairs of dual m wavelet frames and n wavelet frames in l 2 r s 1 and l 2 r s 2 respectively where m and n are s 1xs 1 and s 2xs 2 dilation matrices with a 1 det m 1 and 2a 2 1 det n 1 moreover their mask symbols both satisfy mixed extension principle mep based on the mask symbols a family of nonseparable dual w wavelet frames in l 2 r s are constructed where s s 1 s 2 and w m qon with q and m 1 q both being integer matrices then a convolution scheme for improving regularity of wavelet frames is given from the nonseparable dual w wavelet frames nonseparable w wavelet frames with high regularity can be constructed easily we give an algorithm for constructing nonseparable dual symmetric or antisymmetric wavelet frames in l 2 r s from the dual w wavelet frames nonseparable dual w wavelet frames with symmetry can be obtained easily in the end two examples are given

mazur s conjecture on higher heegner points in this article we establish a non triviality statement for heegner points which was conjectured by b mazur 10 and has subsequently been used as a working hypothesis by a few authors in the study of the arithmetic of elliptic curves

independence of points on elliptic curves arising from special points on modular and shimura curves i global results given a correspondence between a modular curve s and an elliptic curve a we prove that the intersection of any finite rank subgroup of a with the set of points on a corresponding to cm points on s is finite we prove also a version in which s is replaced by a shimura curve and a is replaced by a higher dimensional abelian variety

finite time state estimation for coupled markovian neural networks with sensor nonlinearities this paper investigates the issue of finite time state estimation for coupled markovian neural networks subject to sensor nonlinearities where the markov chain with partially unknown transition probabilities is considered a luenberger type state estimator is proposed based on incomplete measurements and the estimation error system is derived by using the kronecker product by using the lyapunov method sufficient conditions are established which guarantee that the estimation error system is stochastically finite time bounded and stochastically finite time stable respectively then the estimator gains are obtained via solving a set of coupled linear matrix inequalities finally a numerical example is given to illustrate the effectiveness of the proposed new design method

manifold regularized reinforcement learning this paper introduces a novel manifold regularized reinforcement learning scheme for continuous markov decision processes smooth feature representations for value function approximation can be automatically learned using the unsupervised manifold regularization method the learned features are data driven and can be adapted to the geometry of the state space furthermore the scheme provides a direct basis representation extension for novel samples during policy learning and control the performance of the proposed scheme is evaluated on two benchmark control tasks i e the inverted pendulum and the energy storage problem simulation results illustrate the concepts of the proposed scheme and show that it can obtain excellent performance

on a family of orthonormal scalar wavelets and related balanced multiwavelets for a family of length 4n orthonormal scalar wavelet filters that has been shown to be closely related to symmetric antisymmetric orthonormal multifilters saomfs a sufficient and necessary condition is given for their construction a near linear criterion is then provided for constructing balanced multiwavelet filters related to these scalar filters numerical results are presented to demonstrate the potential usefulness of this criterion

probability distribution functions of the grincevicjus series abstract given a sequence n n of independent identically distributed vectors of random variables we consider the grincevicjus series n 1 n k 1 n 1 k and a functional integral equation connected with it we prove that the equation characterizes all probability distribution functions of the grincevicjus series moreover some application of this characterization to a continuous refinement equation is presented

generalized hamilton jacobi bellman formulation based neural network control of affine nonlinear discrete time systems in this paper we consider the use of nonlinear networks towards obtaining nearly optimal solutions to the control of nonlinear discrete time dt systems the method is based on least squares successive approximation solution of the generalized hamilton jacobi bellman ghjb equation which appears in optimization problems successive approximation using the ghjb has not been applied for nonlinear dt systems the proposed recursive method solves the ghjb equation in dt on a well defined region of attraction the definition of ghjb pre hamiltonian function hjb equation and method of updating the control function for the affine nonlinear dt systems under small perturbation assumption are proposed a neural network nn is used to approximate the ghjb solution it is shown that the result is a closed loop control based on an nn that has been tuned a priori in offline mode numerical examples show that for the linear dt system the updated control laws will converge to the optimal control and for nonlinear dt systems the updated control laws will converge to the suboptimal control

high order balanced multiwavelets theory factorization and design this paper deals with multiwavelets and the different properties of approximation and smoothness associated with them in particular we focus on the important issue of the preservation of discrete time polynomial signals by multifilterbanks we introduce and detail the property of balancing for higher degree discrete time polynomial signals and link it to a very natural factorization of the refinement mask of the lowpass synthesis multifilter this factorization turns out to be the counterpart for multiwavelets of the well known zeros at spl pi condition in the usual scalar wavelet framework the property of balancing also proves to be central to the different issues of the preservation of smooth signals by multifilterbanks the approximation power of finitely generated multiresolution analyses and the smoothness of the multiscaling functions and multiwavelets using these new results we describe the construction of a family of orthogonal multiwavelets with symmetries and compact support that is indexed by increasing order of balancing in addition we also detail for any given balancing order the orthogonal multiwavelets with minimum length multifilters

on the convergence of optimistic policy iteration we consider a finite state markov decision problem and establish the convergence of a special case of optimistic policy iteration that involves monte carlo estimation of q values in conjunction with greedy policy selection we provide convergence results for a number of algorithmic variations including one that involves temporal difference learning bootstrapping instead of monte carlo estimation we also indicate some extensions that either fail or are unlikely to go through

solvable points on genus one curves a genus one curve defined over q which has points over qp for all primes p may not have a rational point it is natural to study the classes of q extensions over which all such curves obtain a global point in this article we show that every such genus one curve with semistable jacobian has a point defined over a solvable extension of q

policy iteration adaptive dynamic programming algorithm for discrete time nonlinear systems this paper is concerned with a new discrete time policy iteration adaptive dynamic programming adp method for solving the infinite horizon optimal control problem of nonlinear systems the idea is to use an iterative adp technique to obtain the iterative control law which optimizes the iterative performance index function the main contribution of this paper is to analyze the convergence and stability properties of policy iteration method for discrete time nonlinear systems for the first time it shows that the iterative performance index function is nonincreasingly convergent to the optimal solution of the hamilton jacobi bellman equation it is also proven that any of the iterative control laws can stabilize the nonlinear systems neural networks are used to approximate the performance index function and compute the optimal control law respectively for facilitating the implementation of the iterative adp algorithm where the convergence of the weight matrices is analyzed finally the numerical results and analysis are presented to illustrate the performance of the developed method

balanced multiwavelets theory and design this article deals with multiwavelets which are a generalization of wavelets in the context of time varying filter banks and with their applications to signal processing and especially compression by their inherent structure multiwavelets are fit for processing multichannel signals this is the main issue in which we are interested first we review material on multiwavelets and their links with multifilter banks and especially time varying filter banks then we have a close look at the problems encountered when using multiwavelets in applications and we propose new solutions for the design of multiwavelets filter banks by introducing the so called balanced multiwavelets