the rubin stark conjecture for imaginary abelian fields of odd prime power conductor we build upon ideas developed in 9 as well as results of greither on a strong form of brumer s conjecture 2 4 and prove rubin s integral version of stark s conjecture for imaginary abelian extensions of q of odd prime power conductor

classical solutions for a generalized euler equation in two dimensions abstract it is well known that the euler equations in two spatial dimensions have global classical solutions we provide a new proof which is analytic rather than geometric it is set in an abstract framework that applies to the so called lake and the great lake equations describing weakly non hydrostatic effects of bottom topography on the motion of shallow water the key ingredient is a new lpestimate on the nonlinear term the estimate is used to develop a global hmtheory for bounded domains in r 2which is similar in spirit to a 1975 paper by r temam it also provides explicit bounds on the hmnorm which grow like exp exp t

bounds for the solutions of the complex ginzburg landau equation in terms of the dispersion parameters abstract the diameter in the l norm of the global attractor of the complex ginzburg landau equation ut 1 i u ru 1 i u2 u is estimated by using weighted energy estimates for the solutions on the whole space r d for all parameters d and for which global existence is known we obtain the bound c d r 1 2 upper estimates for c are given which are polynomial in for d

dirichlet inhomogeneous boundary value problem for the n 1 complex ginzburg landau equation abstract we study the following complex ginzburg landau equation with cubic nonlinearity on r n for t 0 u t a i u b i u 2 u a b 0 under initial and dirichlet boundary conditions u x 0 h x for x u x t q x t on where h q are given smooth functions under suitable conditions we prove the existence of a global solution in h1 further this solution approaches to the solution of the nls limit under identical initial and boundary data as a b 0

neumann inhomogeneous boundary value problem for the n 1 complex ginzburg landau equation abstract we study the following neumann inhomogeneous boundary value problem for the complex ginzburg landau equation on r n n 3 u t a i u b i u 2 u a b t 0 under initial condition u x 0 h x for x and neumann boundary condition u n k x t on where h k are given functions under suitable conditions we prove the existence of a global solution in h1

on class groups of random number fields the main aim of the present paper is to disprove the cohen lenstra martinet heuristics in two different ways and to offer possible corrections we also recast the heuristics in terms of arakelov class groups giving an explanation for the probability weights appearing in the general form of the heuristics we conclude by proposing a rigorously formulated cohen lenstra martinet conjecture

stickelberger ideals and fitting ideals of class groups for abelian number fields in this paper we determine completely the initial fitting ideal of the minus part of the ideal class group of an abelian number field over q up to the 2 component this answers an open question of mazur and wiles invent math 76 179 330 1984 up to the 2 component and proves conjecture 0 1 in kurihara j reine angew math 561 39 86 2003 we also study brumer s conjecture and prove a stronger version for a cm field assuming certain conditions in particular on the galois group

on a conjecture of brumer in a series of papers iwasawa proposed and studied analogues for number fields of constructions due to weil for curves over finite fields cf 11 in particular he conjectured an analogue for weil s theorem relating the zeta function of a non singular curve over a finite field with the characteristic polynomial of its frobenius automorphism one strong piece of evidence for this came from the classical theorem of stickelberger however iwasawa s original conjecture was proved for f q in mw without use of stickelberger s theorem for a general totally real number field brumer proposed a generalization of stickelberger s theorem in w2 we proved iwasawa s conjecture for an arbitrary totally real field and here we use this to recover a version of brumer s conjecture we note however that iwasawa s conjecture alone is not sufficient to prove our theorem owing to the phenomenon of trivial zeroes the reason is that in iwasawa theory one habitually interpolates the special values of lfunctions which are imprimitive the key part of this paper is the development of a technique introduced in w2 of adjoining an auxiliary variable to the p adic b functions in a way that permits the study of these primitive lfunctions i am grateful to the referee for correcting some obscure arguments let f be a totally real number field and let k be a cm field which is

numerically derived scalings for the complex ginzburg landau equation abstract we describe some new numerical results concerning the scaling of norms on the turbulent attractor of the quintic complex ginzburg landau equation ut 1 i uxx ru 1 i u u 4 posed on the one dimensional interval 0 1 with periodic boundary conditions the evidence suggests that the real r asymptotic growth rates of some norms are lower than available analytical estimates

a dirichlet inhomogeneous boundary value problem for a generalized ginzburg landau equation abstract the ginzburg landau equation has been used as a simplified mathematical model for various pattern formation systems in mechanics physics and chemistry in this paper we study a generalized complex ginzburg landau equation in two spatial dimension with a fifth order nonlinear term and cubic terms containing spatial derivatives we prove a global existence and uniqueness theorem

existence and decay estimates of solutions to complex ginzburg landau type equations abstract this paper deals with the initial boundary value problem denoted by cgl for the complex ginzburg landau type equation u t i u i u q 1 u u 0 with initial data u 0 l p in the case 1 q 1 2 p n where is bounded or unbounded in r n 0 r there are a lot of studies on local and global existence of solutions to cgl including the physically relevant case q 3 and 0 this paper gives existence results with precise properties of solutions and rigorous proof from a mathematical point of view the physically relevant case can be considered as a special case of the results moreover in the case 0 local and global existence of solutions with the decay estimate u t l p c 1 e c 2 t c 1 c 2 are positive constants is obtained under some conditions the key to the local existence is to construct a semigroup e t i and its l p l q estimate on the other hand the key to the global existence is to derive estimates for solutions by using a kind of interpolation inequality with re v p 2 v i v

control of rare intense events in spatiotemporally chaotic systems we address the problem of using feedback control for the purpose of suppressing rare intense events in spatially extended systems as an example we investigate the use of control to suppress turbulent spikes in the complex ginzburg landau equation in the limit of small dissipation we explore how information obtained by forecasting can be used to implement spatially and temporally localized control parameter changes and how control strength and cost are related to effectiveness in this framework the effects of model error and imperfect state measurement are also considered

the iwasawa conjecture for totally real fields let f be a totally real number field let p be a prime number and for any integer n let fun denote the group of nth roots of unity let 41 be a p adic valued artin character for f and let f be the extension of f attached to 4 i e so that 4 is the character of a faithful representation of gal f f we will assume that f is also totally real for a number field k let k denote the cyclotomic zp extension of k following greenberg we say that 4 is of type s if f n fc f and of type w if 4 is one dimensional with f p c fcc deligne and ribet in dr following kubota and leopoldt for the case f q have proved the existence of a p adic l function associated to a one dimensional artin character 4 with f totally real this function lp s 4 is continuous for s e zp 1 and even at s 1 if 4 is not trivial and satisfies the following interpolation property

weak and strong solutions of the complex ginzburg landau equation abstract the generalized complex ginzburg landau equation t a ra 1 iv a 1 i a 2 a has been proposed and studied as a model for turbulent dynamics in nonlinear partial differential equations it is a particularly interesting model in this respect because it is a dissipative version of the hamiltonian nonlinear schrodinger equation possessing solutions that form localized singularities in finite time in this paper we investigate existence and regularity of solutions to this equation subject to periodic boundary conditions in various spatial dimensions appropriately defined weak solutions are established globally in time and unique strong solutions are found locally a new collection of a priori estimates are presented and we discuss the relationship of our results for the complex ginzburg landau equation to analogous issues for fluid turbulence described by the incompressible navier stokes equations

symmetry and chaos in the complex ginzburg landau equation ii translational symmetries abstract the complex ginzburg landau cgl equation on a one dimensional domain with periodic boundary conditions has a number of different symmetries and solutions of the equation may or may not be fixed by the action of these symmetries we investigate the stability of chaotic solutions that are spatially periodic with period l with respect to subharmonic perturbations that have a spatial period kl for some integer k 1 this is done by considering the isotypic decomposition of the space of solutions and finding the dominant lyapunov exponent associated with each isotypic component we find a region of parameter space in which there exist chaotic solutions with spatial period l and homogeneous neumann boundary conditions that are stable with respect to perturbations of period 2l on varying the parameters it is possible to arrange for this solution to become unstable to perturbations of period 2l while remaining chaotic leading to a supercritical subharmonic blowout bifurcation for a large number of parameter values checked chaotic solutions with spatial period l were found to be unstable with respect to perturbations of period 3l we conclude that while periodic boundary conditions are often convenient mathematically we would not expect to see chaotic spatially periodic solutions forming starting with an arbitrary non periodic initial condition

on the rubin stark conjecture for a special class of cm extensions of totally real number fields we use greither s recent results on brumer s conjecture to prove rubin s integral version of stark s conjecture up to a power of 2 for an infinite class of cm extensions of totally real number fields called nice extensions under the assumption that a certain iwasawa invariant vanishes as a consequence and under the same assumption we show that the brumer stark conjecture is true for nice extensions up to a power of 2

death of period doublings locating the homoclinic doubling cascade abstract this paper studies a natural mechanism called a homoclinic doubling cascade for the disappearance of period doubling cascades in vector fields simply put an entire period doubling cascade collides with a saddle type equilibrium homoclinic doubling cascades are known to have self similar structure in contrast to the well known feigenbaum constant the scaling constants for homoclinic doubling depend on the eigenvalues of the saddle equilibrium specifically we present here for the first time a detailed study of homoclinic doubling cascades in a smooth vector field namely a three dimensional polynomial model proposed by sandstede a numerical algorithm is presented for computing homoclinic doubling cascades in general vector fields which makes use of the program auto homcont this allows us to compute two types of homoclinic doubling cascades one where the primary homoclinic orbit undergoes an inclination flip bifurcation and one where it undergoes an orbit flip bifurcation our results bring out the scaling constants in good agreement with analytical estimates obtained from one dimensional maps

explicit estimates on the torus for the sup norm and the dissipative length scale of solutions of the swift hohenberg equation in one and two space dimensions abstract in this work we have obtained explicit and accurate estimates of the sup norm for solutions of the swift hohenberg equation she in one and two space dimensions by using the best so far available estimates of the embedding constants which appear in the classical functional interpolation inequalities used in the study of solutions of dissipative partial differential equations we have evaluated in an explicit manner the values of the sup norm of the solutions of the she in addition we have calculated the so called time averaged dissipative length scale associated to the above solutions

the brumer stark conjecture in some families of extensions of specified degree as a starting point an important link is established between brumer s conjecture and the brumer stark conjecture which allows one to translate recent progress on the former into new results on the latter for example if k f is an abelian extension of relative degree 2p p an odd prime we prove the l part of the brumer stark conjecture for all odd primes l p with f belonging to a wide class of base fields in the same setting we study the 2 part and p part of brumer stark with no special restriction on f and are left with only two well defined specific classes of extensions that elude proof extensive computations were carried out within these two classes and a complete numerical proof of the brumer stark conjecture was obtained in all cases

on the possibility of soft and hard turbulence in the complex ginzburg landau equation abstract we present analytical methods which predict the occurrence of both soft weak and hard strong turbulence in the complex ginzburg landau cgl equation a t ra 1 i a 1 i a a 2 on a periodic domain 0 1 d in d spatial dimensions hard turbulence is characterised by large fluctuations away from spatial and temporal averages with a cascade of energy to small scales this form of hard turbulence appears to occur not in 1d but only in 2d and 3d in parameter regions which are bounded by hyperbolic curves in the second and fourth quadrants of the planes where the system is modulationally unstable 1 2n f n n 1 a 2 n a 2n dx for n 0 for large times and large r upper bounds exist for the infinite set of fn s constructed from the hierarchy of differential inequalities fn 2nr cn a 2 fn bnf2n fn 1 for cn bn 0 f0 1 estimates for the bottom rung f2 give upper bounds for the whole ladder long time upper bounds on f2 and a 2 and hence all fn are well controlled in the soft region but become much larger in the hard region whereas spati al and temporal averages remain comparatively small when the nonlinearity is a a 2q the critical case qd 2 gives parallel results

numerical solution of calcium mediated dendritic branch model the standard algorithms for spatial discretizations of calcium mediated dendritic branch models via finite difference methods are quite accurate but they are also extremely slow to improve computational efficiency we apply spatial discretization using a spectral collocation method simulations using the spectral collocation method are compared to the finite difference approach using a model for calcium mediated restructuring with spine pruning we find that the spectral collocation method is about fifteen times more efficient to achieve similar accuracy than the finite difference approach even though spectral collocation requires more steps

computing fitting ideals of iwasawa modules this paper determines in an equivariant sense the fitting ideals of several iwasawa modules including the most canonical one the connection between the modules themselves which are usually not of finite projective dimension and the required auxiliary modules of finite projective dimension is made rather explicit the resulting fitting ideals look like stickelberger ideals and there is a close relation with recent work of kurihara as an application we obtain new evidence in favor of the brumer stark conjecture

long range scattering for non linear schrodinger and hartree equations in space dimension n 2 we consider the scattering problem for the non linear schrodinger nls equation with a power interaction with critical powerp 1 2 n in space dimensionsn 2 and 3 and for the hartree equation with potential x 1 in space dimensionn 2 we prove the existence of modified wave operators in thel2 sense on a dense set of small and sufficiently regular asymptotic states

intermittency and regularity issues in 3d navier stokes turbulence two related open problems in the theory of 3d navier stokes turbulence are discussed in this paper the first is the phenomenon of intermittency in the dissipation field dissipation range intermittency was first discovered experimentally by batchelor and townsend over fifty years ago it is characterized by spatio temporal binary behaviour in which long quiescent periods in the velocity signal are interrupted by short active events during which there are violent fluctuations away from the average the second and related problem is whether solutions of the 3d navier stokes equations develop finite time singularities during these events this paper shows that leray s weak solutions of the three dimensional incompressible navier stokes equations can have a binary character in time the time axis is split into good and bad intervals on the good intervals solutions are bounded and regular whereas singularities are still possible within the bad intervals an estimate for the width of the latter is very small and decreases with increasing reynolds number it also decreases relative to the lengths of the good intervals as the reynolds number increases within these bad intervals lower bounds on the local energy dissipation rate and other quantities such as u t and u t are very large resulting in strong dynamics at sub kolmogorov scales intersections of bad intervals for n 1 are related to the potentially singular set in time it is also proved that the navier stokes equations are conditionally regular provided in a given bad interval the energy has a lower bound that is decaying exponentially in time

asymptotics for the generalized two dimensional ginzburg landau equation abstract in this paper the authors have studied a generalized ginzburg landau equation in two spatial dimensions 2d they have shown that this equation under periodic boundary conditions has the maximal attractor with finite hausdorff dimension this rigorously establishes the foundation for further investigation of this type of model

global existence of solutions to the derivative 2d ginzburg landau equation abstract in this paper we study a complex derivative ginzburg landau equation with two spatial variables 2d we obtain sufficient conditions for the existence and uniqueness of global solutions for the initial boundary value problem of the derivative 2d ginzburg landau equation and improve the known results

the time periodic solution to a 2d generalized ginzburg landau equation abstract in this paper we study a 2d generalized ginzburg landau equation with a periodic boundary condition the existence and uniqueness of a time periodic solution to this equation is proved

minus class groups of the fields of the l th roots of unity we show that for any prime number l 2 the minus class group of the field of the l th roots of unity q p l admits a finite free resolution of length 1 as a module over the ring z g 1 i here i denotes complex conjugation in g gal q p l q p z lz moreover for the primes l 509 we show that the minus class group is cyclic as a module over this ring for these primes we also determine the structure of the minus class group

traveling waves in the baer and rinzel model of spine studded dendritic tissue abstract the baer and rinzel model of dendritic spines uniformly distributed along a dendritic cable is shown to admit a variety of regular traveling wave solutions including solitary pulses multiple pulses and periodic waves we investigate numerically the speed of these waves and their propagation failure as functions of the system parameters by numerical continuation multiple pulse waves are shown to occur close to the primary pulse except in certain exceptional regions of parameter space which we identify the propagation failure of solitary and multiple pulse waves is shown to be associated with the destruction of a saddle node bifurcation of periodic orbits the system also supports many types of irregular wave trains these include waves which may be regarded as connections to periodics and bursting patterns in which pulses can cluster together in well defined packets the behavior and properties of both these irregular spike trains is explained within a kinematic framework that is based on the times of wave pulses the dispersion curve for periodic waves is important for such a description and is obtained in a straightforward manner using the numerical scheme developed for the study of the speed of a periodic wave stability of periodic waves within the kinematic theory is given in terms of the derivative of the dispersion curve and provides a weak form of stability that may be applied to solutions of the traveling wave equations the kinematic theory correctly predicts the conditions for period doubling bifurcations and the generation of bursting states moreover it also accurately describes the shape and speed of the traveling front that connects waves with two different periods

localized estimates and cauchy problem for the logarithmic complex ginzburg landau equation we prove that the localization of the l2 and similar estimates for the complex ginzburg landau equation 1 known to hold in the power case g also holds in the logarithmic case g log for large with 2 as a consequence the theory of the cauchy problem in local spaces given in a previous paper also extends to the logarithmic case

remarks on the breakdown of smooth solutions for the 3 d euler equations the authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3 d euler equations in other words if a solution of the euler equations is initially smooth and loses its regularity at some later time then the maximum vorticity necessarily grows without bound as the critical time approaches equivalently if the vorticity remains bounded a smooth solution persists

on the equivariant tamagawa number conjecture in tame cm extensions let l k be a finite galois cm extension with galois group g the equivariant tamagawa number conjecture etnc for the pair h 0 rm spec l 0 mathbb z g naturally decomposes into p parts where p runs over all rational primes if p is odd these p parts in turn decompose into a plus and a minus part let l k be tame above p we show that a certain ray class group of l defines an element in k 0 mathbb z p g mathbb q p which is determined by a corresponding stickelberger element if and only if the minus part of the etnc at p holds for this we use the lifted root number conjecture for small sets of places which is equivalent to the etnc in the number field case for abelian g we show that the minus part of the etnc at p implies the strong brumer stark conjecture at p we prove the minus part of the etnc at p for almost all primes p

on the dimension of the attractors in two dimensional turbulence abstract using a new version of the sobolev lieb thirring inequality we derive an upper bound for the dimension of the universal attractor for two dimensional space periodic navier stokes equations this estimate is optimal up to a logarithmic correction the relevance of this estimate to turbulence and related results are also briefly discussed

length scales in solutions of the complex ginzburg landau equation abstract we generalise and in certain cases improve on previous a priori estimates of sobolev norms of solutions to the generalised complex ginzburg landau equation a set of dynamic length scales based on ratios of these norms is defined we are able to derive lower bounds for time averages and long time limits of these length scales the bounds scale like the inverses of our l bounds

integrality of stickelberger elements and the equivariant tamagawa number conjecture let l k be a nite galois cm extension of number elds with galois group g in an earlier paper the author has de ned a module sku l k over the center of the group ring zg which coincides with the sinnott kurihara ideal if g is abelian and in particular contains many stickelberger elements it was shown that a certain conjecture on the integrality of sku l k implies the minus part of the equivariant tamagawa number conjecture at an odd prime p for an in nite class of non abelian galois cm extensions of number elds which are at most tamely rami ed above p provided that iwasawa s invariant vanishes here we prove a relevant part of this integrality conjecture which enables us to deduce the equivariant tamagawa number conjecture from the vanishing of for the same class of extensions

the sil nikov problem exponential expansion strong λ lemma c1 linearization and homoclinic bifurcation on introduit une nouvelle theorie appelee le developpement exponentiel et on explore ses interactions avec la theorie de la variete invariante locale les lemmes et la theorie de linerarisation reguliere

space time invariant measures entropy and dimension for stochastic ginzburg landau equations we consider a randomly forced ginzburg landau equation on an unbounded domain the forcing is smooth and homogeneous in space and white noise in time we prove existence and smoothness of solutions existence of an invariant measure for the corresponding markov process and we define the spatial densities of topological entropy of measure theoretic entropy and of upper box counting dimension we prove inequalities relating these different quantities the proof of existence of an invariant measure uses the compact embedding of some space of uniformly smooth functions into the space of locally square integrable functions and a priori bounds on the semi flow in these spaces the bounds on the entropy follow from spatially localised estimates on the rate of divergence of nearby orbits and on the smoothing effect of the evolution