towards hilbert s tenth problem for rings of integers through iwasawa theory and heegner points for a positive proportion of primes p and q we prove that mathbb z is diophantine in the ring of integers of mathbb q root 3 of p sqrt q this provides a new and explicit infinite family of number fields k such that hilbert s tenth problem for o k is unsolvable our methods use iwasawa theory and congruences of heegner points in order to obtain suitable rank stability properties for elliptic curves

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

new numerical algorithms for the nonlinear diffusion model of image denoising and segmentation abstract in this paper for the nonlinear diffusion model of image denoising and segmentation we present two high order explicit algorithms and a semi implicit algorithm using algebraic multi grid method the unconditional stability of the semi implicit approximation is given the numerical results of the above three algorithms are demonstrated with comparison to general explicit algorithm in y chen b c vemuri l wang image denoising and segmentation via nonlinear diffusion comput math appl 39 2000 131 149 for real noised images the numerical results show that the implicit algorithm is quicker and sharper than all explicit algorithms

klein forms and the generalized superelliptic equation if f x y 2 z x y is an irreducible binary form of degree k 3 then a theorem of darmon and granville implies that the generalized superelliptic equation f x y z l has given an integer l maxf2 7 kg at most nitely many solutions in coprime integers x y and z in this paper for large classes of forms of degree k 3 4 6 and 12 including heuristically most cubic forms we extend this to prove a like result where the parameter l is now taken to be variable in the case of irreducible cubic forms this provides the rst examples where such a conclusion has been proven the method of proof combines classical invariant theory modular galois representations and properties of elliptic curves with isomorphic mod n galois representations

design of robust controllers for time delay systems the problem of stabilizing linear dynamical time delay systems subject to bounded uncertainties is investigated two memoryless feedback controllers are considered it is established that when the matching conditions are met and certain bounding relations are satisfied then the linear controller renders the zero response of the system asymptotically stable saturation type controllers are shown to guarantee that all system responses are uniformly ultimately bounded

continuous feedback guaranteeing uniform ultimate boundedness for uncertain linear delay systems an application to river pollution control we employ the stability result of thowsen on a class of linear delay systems containing uncertain elements and subject to uncertain inputs to manage a simple one reach model of a river system with time delay and subject to uncertain but bounded input as well as parameter disturbances two important indices of water quality namely biochemical oxygen demand bod and dissolved oxygen do are used as state variables while controls in the form of effluent discharge rate of bod and in stream aeration rate of do are employed to steer the system to a calculable neighborhood of a desired steady state and then maintain it there

an improved razumikhin type theorem and its applications an improved razumikhin type theorem is established and the advantage of the established theorem for reducing conservatism is discussed for a class of time delay systems a simple asymptotic stability condition is deduced as an application of the established theorem two illustrative examples are given to show the superiority of the results to those in the literature

filtering on sampled data systems with parametric uncertainty the paper is concerned with the problem of robust h sub spl infin filtering for a class of systems with parametric uncertainties and unknown time delays under sampled measurements the parameter uncertainties considered are real time varying and norm bounded appearing in the state equation an approach has been proposed for the designing of h sub spl infin filters using sampled measurements which would guarantee a prescribed h sub spl infin performance in the continuous time context irrespective of the parameter uncertainties and unknown time delays both cases of finite and infinite horizon filtering are studied it has been shown that the above robust h sub spl infin filtering problem can be solved in terms of differential riccati inequalities with finite discrete jumps

modular elliptic curves and fermat s last theorem when andrew john wiles was 10 years old he read eric temple bell s the last problem and was so impressed by it that he decided that he would be the first person to prove fermat s last theorem this theorem states that there are no nonzero integers a b c n with n 2 such that an bn cn the object of this paper is to prove that all semistable elliptic curves over the set of rational numbers are modular fermat s last theorem follows as a corollary by virtue of previous work by frey serre and ribet

dynamical feedback compensator for uncertain time delay systems containing saturating actuator abstract the problem of the stabilization of an uncertain time delay system with a saturating actuator is discussed these uncertainties may be linear non linear and or time varying but only the upper bounds are known a linear dynamical feedback compensator is used to stabilize this uncertain saturating and time delay system the concept of matrix measure and the comparison theory are employed to investigate the robust stability condition which assures asymptotic stability one example is given to illustrate the application of the proposed design method

second p descents on elliptic curves let p be a prime and let c be a genus one curve over a number field k representing an element of order dividing p in the shafarevich tate group of its jacobian we describe an algorithm which computes the set of d in the shafarevich tate group such that pd c and obtains explicit models for these d as curves in projective space this leads to a practical algorithm for performing explicit 9 descents on elliptic curves over q

on the modularity of elliptic curves over q wild 3 adic exercises in this paper building on work of wiles wi and of wiles and one of us r t tw we will prove the following two theorems see 2 2 theorem a if e q is an elliptic curve then e is modular theorem b if gal q q gl2 f5 is an irreducible continuous representation with cyclotomic determinant then is modular we will first remind the reader of the content of these results and then briefly outline the method of proof if n is a positive integer then we let 1 n denote the subgroup of sl2 z consisting of matrices that modulo n are of the form 1 0 1

a lagrange approach to set theoretic control synthesis a lagrange approach to solving the nonlinear constrained optimization problem arising in the set theoretic control problem is described by introducing matrix lagrange multipliers the problem is reduced to that of solving a set of nonlinear simultaneous matrix equations one of which is the familiar matrix riccati equation frequently encountered in linear quadratic control theory the structural similarities and differences between set theoretic and linear quadratic control methods are identified the results obtained from the set theoretic control approach are compared with those obtained from the linear quadratic control approach

the fontaine mazur conjecture for gl _2 introduction 641 1 breuil mezard conjecture and the p adic local langlands 644 1 1 the breuil mezard conjecture 644 1 2 review of colmez s functor 647 1 3 hilbert samuel multiplicities 650 1 4 representations and pseudo representations 653 1 5 gl2 qp representations mod p 657 1 6 local patching and multiplicities 661 1 7 from pseudo representations to representations 669 2 modularity via the breuil mezard conjecture 675 2 1 quaternionic forms 675 2 2 global patching and multiplicities 677 2 3 the breuil mezard conjecture 686 addendum to ki 1 687 references 689

technical communique decay estimates for applications of razumikhin type theorems decay estimates are established for a family of razumikhin type retarded functional differential equations rfdes from the point of applications of razumikhin type theorems the work presented in this paper is an extension of the ones reported so far in the related literature since many contributions are based on razumikhin type theorems our results may provide effective methods for quantitative stability analysis of delayed systems encountered in engineering

quadratic stabilization of continuous time systems with state delay and norm bounded time varying uncertainties this paper develops a method for designing a feedback control law to stabilize a class of uncertain linear systems the class of systems under consideration is described by a continuous time model with variable state delay and depends on time varying unknown but bounded uncertain parameters it is shown that the construction of the stabilizing controller involves solving a certain algebraic riccati equation several previous results are derived as interesting special cases finally a simulation example is presented to illustrate the analytical developments

the local root number of elliptic curves with wild ramification let e be an elliptic curve over a number field f the root number is conjecturally the sign of the functional equation of l function of e f it is defined as the product of local signs over all places of f the purpose of this paper is to describe this local sign by the coefficients of a weierstras equation of e

variable exponent linear growth functionals in image restoration we study a functional with variable exponent 1 leq p x leq 2 which provides a model for image denoising enhancement and restoration the diffusion resulting from the proposed model is a combination of total variation tv based regularization and gaussian smoothing the existence uniqueness and long time behavior of the proposed model are established experimental results illustrate the effectiveness of the model in image restoration

continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems we consider a dynamic system containing uncertain elements only the set of possible values of these uncertainties is known based on this information a class of state feedback controls is proposed in order to guarantee uniform ultimate boundedness of every system response within an arbitrarily small neighborhood of the zero state these feedback controls are continuous functions of the state

memoryless stabilization of uncertain dynamic delay systems riccati equation approach the authors present a procedure for obtaining the memoryless linear state feedback control of uncertain dynamic delay systems the uncertainties are time varying and within a given compact set this method is an extension of the riccati equation approach proposed by i r petersen and c v hollot 1986 the extension is straightforward also the uncertainties do not need to satisfy the matching conditions

a nonlinear primal dual method for total variation based image restoration we present a new method for solving total variation tv minimization problems in image restoration the main idea is to remove some of the singularity caused by the nondifferentiability of the quantity nabla u in the definition of the tv norm before we apply a linearization technique such as newton s method this is accomplished by introducing an additional variable for the flux quantity appearing in the gradient of the objective function which can be interpreted as the normal vector to the level sets of the image u our method can be viewed as a primal dual method as proposed by conn and overton a primal dual interior point method for minimizing a sum of euclidean norms preprint 1994 and andersen ph d thesis odense university denmark 1995 for the minimization of a sum of euclidean norms in addition to possessing local quadratic convergence experimental results show that the new method seems to be globally convergent

one approach to the control of uncertain dynamical systems we present a controller design methodology for uncertain systems which is based on the constructive use of lyapunov stability theory the uncertainties which are deterministic are characterized by certain structural conditions and known as well as unknown bounds as a consequence of the lyapunov approach the methodology is not restricted to linear or time invariant systems the robustness of these controllers in the presence of singular perturbations is considered the situation in which the full state of the system is not available for measurement is also considered as are other generalizations

a stabilizing control for a class of uncertain linear delayed systems via state feedback development of the results abstract stabilization of uncertain delayed systems via linear state feedback including no delays is considered the present report is a development of the previous results it is shown that the obtained condition is fundamentally equivalent to wei s results called antisymmetric stepwise configuration he constructed his results for the stabilization of a linear system including no delays

fourth order partial differential equations for image enhancement abstract second order partial differential equations have been studied as a useful tool for noise removal the perona malik model p perona malik scale space and edge detection using anisotropic diffusion ieee trans pattern anal mach intell 12 1990 629 639 has an edge preserving property but have sometimes the undesirable blurred effect in this paper we propose improved models by combining catte et al s model f catte p l lions j m morel t coll image selective smoothing and edge detection by nonlinear diffusion siam j numer anal 129 1992 182 193 with fourth order terms we prove the existence and uniqueness of the proposed models then we show numerical evidence of the power of resolution of these models with respect to other known models as the perona malik model the catte et al s model the modified total variation model by chan et al t chan a marquina p mulet second order differential functionals in total variation based image restoration available from etc

noise removal using fourth order partial differential equation with applications to medical magnetic resonance images in space and time we introduce a new method for image smoothing based on a fourth order pde model the method is tested on a broad range of real medical magnetic resonance images both in space and time as well as on nonmedical synthesized test images our algorithm demonstrates good noise suppression without destruction of important anatomical or functional detail even at poor signal to noise ratio we have also compared our method with related pde models

convergence of a fixed point iteration method for the osv model in image processing image denoising and texture extraction are important problems in which many new methods recently have been developed one of the most important models is the osv model s osher a sole l vese image decomposition and restoration using total variation minimization and the h 1 norm multiscale model simul a siam interdisciplinary j 1 3 2003 349 370 which is constructed by the total variation and h 1 norm this paper proves the existence of the minimizer of the functional from the osv model and analyzes the convergence of an iterative method for solving the problems our iteration method is constructed by a fixed point iteration on the fourth order partial differential equation from the computation of the associated euler lagrange equation and the limit of our iterations satisfies the minimizer of the functional from the osv model in numerical experiments we compare the numerical results of our works with those of the rof model l i rudin s osher e fatemi nonlinear total variation based noise removal algorithms phys d 60 1992 259 268

fourth order partial differential equations for noise removal a class of fourth order partial differential equations pdes are proposed to optimize the trade off between noise removal and edge preservation the time evolution of these pdes seeks to minimize a cost functional which is an increasing function of the absolute value of the laplacian of the image intensity function since the laplacian of an image at a pixel is zero if the image is planar in its neighborhood these pdes attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image piecewise planar images look more natural than step images which anisotropic diffusion second order pdes uses to approximate an observed image so the proposed pdes are able to avoid the blocky effects widely seen in images processed by anisotropic diffusion while achieving the degree of noise removal and edge preservation comparable to anisotropic diffusion although both approaches seem to be comparable in removing speckles in the observed images speckles are more visible in images processed by the proposed pdes because piecewise planar images are less likely to mask speckles than step images and anisotropic diffusion tends to generate multiple false edges speckles can be easily removed by simple algorithms such as the one presented in this paper

memoryless stabilization of uncertain dynamic systems with time varying delayed states and controls we consider the feedback stabilization of uncertain dynamic systems with time varying delays in both states and controls we extend the so called riccati equation approach to include those systems and we give an example

some remarks on the two variable main conjecture of iwasawa theory for elliptic curves without complex multiplication we establish several results towards the two variable main conjecture of iwasawa theory for elliptic curves without complex multiplication over imaginary quadratic fields namely i the existence of an appropriate p adic l function building on works of hida and perrin riou ii the basic structure theory of the dual selmer group following works of coates hachimori venjakob et al and iii the implications of dihedral or anticyclotomic main conjectures with basechange the result of i is deduced from the construction of hida and perrin riou which in particular is seen to give a bounded distribution the result of ii allows us to deduce a corank formula for the p primary part of the tate shafarevich group of an elliptic curve in the z p 2 extension of an imaginary quadratic field finally iii allows us to deduce a criterion for one divisibility of the two variable main conjecture in terms of specializations to cyclotomic characters following a suggestion of greenberg as well as a refinement via basechange c 2011 elsevier inc all rights reserved

average ranks of elliptic curves tension between data and conjecture rational points on elliptic curves are the gems of arithmetic they are to diophantine geometry what units in rings of integers are to algebraic number theory what algebraic cycles are to algebraic geometry a rational point in just the right context at one place in the theory can inhibit and control thanks to ideas of kolyvagin kol88 the existence of rational points and other mathematical structures elsewhere despite all that we know about these objects the initial mystery and excitement that drew mathematicians to this arena in the first place remains in full force today we have a network of heuristics and conjectures regarding rational points and we have massive data accumulated to exhibit instances of the phenomena generally we would expect that our data support our conjectures and if not we lose faith in our conjectures but here there is a somewhat more surprising interrelation between data and conjecture they are not exactly in open conflict one with the other but they are no great comfort to each other either we discuss various aspects of this story including recent heuristics and data that attempt to resolve this mystery we shall try to convince the reader that despite seeming discrepancy data and conjecture are in fact in harmony

finiteness results for modular curves of genus at least 2 a curve x over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i is modular if it is dominated by x 1 n for some n if in addition the image of its jacobian in j 1 n is contained in the new subvariety of j 1 n then x is called a new modular curve we prove that for each g 2 the set of new modular curves over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i of genus g is finite and computable for the computability result we prove an algorithmic version of the de franchis severi theorem similar finiteness results are proved for new modular curves of bounded gonality for new modular curves whose jacobian is a quotient of j 0 n new with n divisible by a prescribed prime and for modular curves new or not with levels in a restricted set we study new modular hyperelliptic curves in detail in particular we find all new modular curves of genus 2 explicitly and construct what might be the complete list of all new modular hyperelliptic curves of all genera finally we prove that for each field k of characteristic zero and g 2 the set of genus g curves over k dominated by a fermat curve is finite and computable

nonlinear multigrid method for solving the llt model our goal in this paper is to study a nonlinear multigrid method for solving the llt model in order to improve the convergence rate of the proposed multigrid method an improved dual iteration is proposed as its smoother furthermore we give the local fourier analysis lfas of the chambolle s dual iterations and a modified smoother for solving the llt model respectively numerical results illustrate the efficiency of the proposed method and indicate that such a multigrid method is more suitable to deal with large sized images

galois sections for abelianized fundamental groups given a smooth projective curve x of genus at least 2 over a number field k grothendieck s section conjecture predicts that the canonical projection from the etale fundamental group of x onto the absolute galois group of k has a section if and only if the curve has a rational point we show that there exist curves where the above map has a section over each completion of k but not over k in the appendix victor flynn gives explicit examples in genus 2 our result is a consequence of a more general investigation of the existence of sections for the projection of the etale fundamental group with abelianized geometric part onto the galois group we also point out the relation to the elementary obstruction of colliot thelene and sansuc

on an estimate of the decay rate for applications of razumikhin type theorems a decay estimate is established for solutions of a class of razumikhin type functional differential equations fdes the fdes investigated here are general enough to satisfy the requirement of applications of razumikhin type theorems with the help of an application of our result it is illustrated that the method presented in this paper is adaptable to quantitative stability analysis of the retarded functional differential equations encountered in engineering

memoryless stabilization of uncertain linear systems including time varying state delays the problem of stabilizing a class of uncertain time delay systems via memoryless linear feedback is examined the systems under consideration are linear systems with time varying state delays they also contain uncertain parameters possibly time varying whose values are known only to within a prescribed compact bounding set the main contribution given is to enlarge the class of time delay systems for which one can construct a stabilizing memoryless linear feedback controller within this framework a novel notion of robust memoryless stabilizability is first introduced via the method of lyapunov functionals then a sufficient condition for the stabilizability is proposed it is shown that solvability of a parameterized riccati equation can be used to determine whether the time delay system satisfies the sufficient condition if there exists a positive definite symmetric solution satisfying the riccati equation a suitable memoryless linear feedback law can be derived

modularity of certain potentially barsotti tate galois representations where is the reduction of p these results were subject to hypotheses on the local behavior of p at x i e the restriction of p to a decomposition group at x and to irreducibility hypotheses on 1p in this paper we build on the methods of 461 451 and 121 and relax the hypotheses on local behavior in particular we treat certain adic representations which are not semistable at x but potentially semistable we do this using results of 61 generalizing a theorem of ramakrishna 321 see fontaine mazur 22 131 for a slightly different point of view the results in 61 show that certain potentially barsotti tate deformation problems are smooth allowing us to define certain universal deformations for with the necessary galoistheoretic properties to apply wiles method to carry out the proof that these deformations are indeed realized in the cohomology of modular curves i e that the universal deformation rings are hecke algebras we need to identify the corresponding cohomology groups and prove they have the modular theoretic properties needed to apply wiles method as in 151 and 121 the identification is made by matching local behavior of automorphic representations and galois representations via the local langlands correspondence together with fontaine s theory at the prime we work directly with cohomology of modular curves instead of

on high order denoising models and fast algorithms for vector valued images variational techniques for gray scale image denoising have been deeply investigated for many years however little research has been done for the vector valued denoising case and the very few existent works are all based on total variation regularization it is known that total variation models for denoising gray scaled images suffer from staircasing effect and there is no reason to suggest this effect is not transported into the vector valued models high order models on the contrary do not present staircasing in this paper we introduce three high order and curvature based denoising models for vector valued images their properties are analyzed and a fast multigrid algorithm for the numerical solution is provided ams subject classifications 68u10 65f10 65k10

uniform ultimate boundedness of the solutions of uncertain dynamic delay systems with state dependent and memoryless feedback control a continuous feedback controller is developed for a class of non linear time varying uncertain systems governed by functional differential equations with retarded arguments the form of the controller guarantees that under certain assumptions all solutions are uniformly bounded in norm for time t sufficiently largo in spite of the uncertainty in system parameters and input signals furthermore given any 0 the parameters in the description of the controller can be chosen such that the closed loop solutions all satisfy x t for large t

smith theory and hecke operators abstract of the connections between the cohomology of arithmetic groups and representations of galois groups some are known and more are conjectured this paper proves one of these conjectures for certain monomial galois representations let p be a prime we show that the representation of the absolute galois group of q induced from an f p valued ray class character of the cyclotomic field q p is attached to a hecke eigenclass in the mod p cohomology of a torsion free congruence subgroup m of gl p 1 z the method involves constructing an action of the hecke algebra on the fundamental exact sequence of smith theory arising from the action by conjugation of an element of order p in gl p 1 z on m

some remarks on landau siegel zeros in this paper we show that under the assumption that all the zeros of the l functions under consideration are either real or lie on the critical line one may considerably improve on the known results on landau siegel zeros

on the delay independent stabilization of linear neutral systems with limited measurable state variables abstract the delay independent stabilizability of linear neutral systems with limited measurable state variables is considered a sufficient condition for the delay independent stabilizability of linear retarded systems with limited measurable state variables and linear neutral systems with measurable state variables which had been obtained by amemiya and maet al is proved to be also valid for linear neutral systems with limited measurable state variables

ring theoretic properties of certain hecke algebras the purpose of this article is to provide a key ingredient of w2 by establishing that certain minimal hecke algebras considered there are complete intersections as is recorded in w2 a method going back to mazur m allows one to show that these algebras are gorenstein but for the complete intersection property a new approach is required the methods of this paper are related to those of chapter 3 of w2 the methods of section 3 of this paper are based on a previous approach of one of us a w we would like to thank henri darmon fred diamond and gerd faltings for carefully reading the first version of this article gerd faltings has also suggested a simplification of our argument as well as of the argument of chapter 3 of w2 and we would like to thank him for allowing us to reproduce these in the appendix to this paper r t would like to thank a w for his invitation to collaborate and for sharing his many insights into the questions considered r t would also like to thank princeton university universite de paris 7 and harvard university for their hospitality during this collaboration a w was supported by an nsf grant

on a conjecture of conrad diamond and taylor we prove a conjecture of conrad diamond and taylor on the size of certain deformation rings parametrizing potentially barsotti tate ga lois representations to achieve this we extend results of breuil and m ezard classifying galois lattices in semistable representations in terms of strongly divisible modules to the potentially crystalline case in hodge tate weights 0 1 we then use these strongly divisible modules to compute the desired deformation rings as a corollary we obtain new results on the modularity of potentially barsotti tate representations

notes on anomalies elliptic curves and the bs d conjecture we consider anomaly cancellation for gauge theories where the left handed chiral multiplets are in higher representations in particular if the left handed quarks and leptons transform under the triplet representation of and if the gauge group is compact then up to an overall scaling there is only one possible nontrivial assignment for the hypercharges if n 3 and two if n 9 otherwise there are infinitely many we use the mordell weil theorem mazur s theorem and the cremona elliptic curve database which uses kolyvagin s theorem on the birch swinnerton dyer conjecture to prove these statements

on ultimate boundedness control of uncertain systems in the absence of matching assumptions this paper describes an approach to ultimate boundedness control under hypotheses less stringent than heretofore required instead of imposing the so called matching conditions of previous authors a certain decomposition of the dynamics is performed the resulting decomposed system has two parts a matched portion and a mismatched portion a certain measure of mismatch m is defined and it is shown that effective control is possible as long as m does not exceed some critical mismatch threshold m it is seen that this threshold depends on the choice of feedback hence it becomes possible to investigate the tradeoffs between the size of the mismatch threshold and the sizes of the feedback gains which achieve the desired ultimate boundedness of solutions