multiplicity of time periodic solutions and their stabilities for quasigeostrophic equations in this article we study the quasigeostrophic equation which is a prototypical geophysical fluid model we will show the existence of multiple time periodic solutions and study their stabilities we also prove the exponential decaying of solutions to the initial value problem with small initial data

the sq universality and residual properties of relatively hyperbolic groups in this paper we study residual properties of relatively hyperbolic groups in particular we show that if a group g is non elementary and hyperbolic relative to a collection of proper subgroups then g is sq universal

bifurcating periodic solutions of wind driven circulation equations the existence of bifurcating periodic flows in a quasi geostrophic mathematical model of wind driven circulation is investigated in the model the ekman number r and reynolds number r control the stability of the motion of the fluid through rigorous analysis it is proved that when the basic steady state solution is independent of the ekman number then a spectral simplicity condition is sufficient to ensure the existence of periodic solutions branching off the basic steady state solution as the ekman number varies across its critical value for constant reynolds number when the basic solution is a function of ekman number an additional condition is required to ensure periodic solutions

behaviour of the frobenius map in a noncommutative world abstract we study various properties of the frobenius map x x p on a noncommutative algebra r over a field f of positive characteristic p we call r perfect whenever is surjective more generally we say that r is additively respectively multiplicatively p power closed if for every x y r there exists a positive integer n such that x p n y p n respectively x p n y p n is a p power in r if for example n were a ring homomorphism as when r is commutative then r would be additively and multiplicatively p power closed we show that a finite dimensional unital algebra r is lie nilpotent if and only if r is additively respectively multiplicatively p power closed and each of its simple homomorphic images has schur index 1 since not all finite dimensional perfect division algebras are lie nilpotent the schur index condition cannot be omitted we deduce that similar results hold for the class of all finitely generated pi algebras moreover for this same class we give a positive solution to the following problem reminiscent of the problems of kurosh and levitzki is every finitely generated perfect algebra finite dimensional

structural classification and stability of divergence free vector fields abstract it is proved that a divergence free vector field v on a 2d submanifold of s2 is structurally stable with divergence free vector field perturbations if and only if 1 v is regular 2 all interior saddle points of v are self connected and 3 each boundary saddle point is connected to boundary saddles on the same connected component of the boundary the main motivation of this article and the companying articles indiana univ math j 50 1 2001 159 appl math lett 12 1999 39 am math soc 238 1999 193 dynamics of 2d incompressible flows in p bates k lu d xu eds differential equations and computational simulations world scientific singapore 2000 pp 270 276 math model numer anal 34 2 2000 419 nonlin anal real world appl 2 2001 467 discrete contin dyn syst 7 2 2001 431 discrete contin dyn syst ser b 1 1 2001 29 is to develop a geometric theory of 2d incompressible fluid flows in the physical spaces

on artin s theorem and azumaya algebras a short proof of the existence of central polynomial of matrix rings is given and some of its applications the following characterization of azumaya algebras is proved r is azumaya iff there exist ai bi r i 1 k with ik 1 airbi z r r r and aibi 1 this is proved as a consequence of the following generalization of a theorem due to m artin and generalized by c procesi let r x1 xk be a p i ring a central noetherian subring then r is azumaya iff for every two sided ideal i in r z ri z r i z r where z r denotes the center of r

attractors for the 3d baroclinic quasi geostrophic equations of large scale atmosphere abstract the existence of absorbing sets and a compact connected global attractor to the 3d baroclinic quasi geostrophic equations of large scale atmosphere are obtained using cft theory an upper bound of the hausdorff and fractal dimensions of the global attractor in terms of the physically relevant parameters re the reynolds number ro the rossby number and f is derived moreover similar results are also established for the 2d vorticity equations of globe atmosphere

asymptotic convergence rate of supercritical surface quasi geostrophic equation in morrey space abstract in this paper the asymptotic convergence rate of the supercritical surface quasi geostrophic equation in morrey space is investigated it is shown that if the global solution x t of the supercritical surface quasi geostrophic equation satisfies the growth condition in critical morrey space 0 t m p q r 2 r d t for r 2 p 2 p then every perturbed weak solution x t of the perturbed quasi geostrophic equation converges asymptotically to x t with the optimal rate t t l 2 o 1 t 1 as t in particular the initial perturbation is allowed large the findings are based on the spectral decomposition methods of fractional laplace and morrey space technique

complete proper minimal surfaces in convex bodies of mathbbr 3 consider a convex domain b of r 3 we prove that there exist complete minimal surfaces which are properly immersed in b we also demonstrate that if d and dare convex domains with d bounded and the closure of d contained in dthen any minimal disk whose boundary lies in the boundary of d can be approximated in any compact subdomain of d by a complete minimal disk which is proper in d we apply these results to study the so called type problem for a minimal surface we demonstrate that the interior of any convex region of r 3 is not a universal region for minimal surfaces in the sense explained by meeks and perez in 9

engel varieties of associative rings and the number of mersenne primes abstract an associative ring r can be viewed as a semigroup via a b a b ab and as a lie ring via a b ab ba it is known that the lie ring r is nilpotent if and only if the adjoint semigroup r is nilpotent in the sense defined by mal cev or by neumann and taylor we prove a similar result for associative rings whose lie rings satisfy an engel identity mersenne primes appear in an unexpected role

on gelfand kirillov dimension and related topics recall that in ii one always has gk m 2 max gk n gk e if equality holds for all short exact sequences of s modules then gk dimension is said to be exact for s modules in general exactness fails quite drastically even in situations which are otherwise considered to be well behaved for example g bergman 3 has constructed an afline pialgebra s having an ideal i of square 0 such that i is cyclic as right ideal of s yet gk s 3 gk s 1 2 thus quite stringent conditions have to be imposed on the algebra or the modules in question for equality to hold in ii on the other hand the integrality question i presumably has a positive answer for many classes of algebras and modules that are of interest although the gk dimension of an algebra module can be 0 1 co or any real number 2 2 0 co or any real number 2 1 warfield 17 it tends to be an integer or co in most cases which arise naturally thus no example of an affrne algebra s which is either noetherian or finitely presented but has finite non integral gk dimension seems to be known however positive results are rare some exceptions where gk dimension is known to be an integer or co are finitely presented monomial algebras govorov 9 almost commutative algebras tauvel 16 an noetherian pi algebras lorenz and small 131 gk dimension by its definition measures the rate of growth of the steps in certain canonically defined filtrations on algebras and modules often

equilibrium states of the charney devore quasi geostrophic equation in mid latitude atmosphere on the predictability of atmospheric blocking in the mid latitude atmosphere a quasi geostrophic beta plane channel model equation was introduced in a pioneering work by charney and devore in 1979 based on the understanding of quasi stationary weather patterns the equation is driven by a zonal thermal flow representing a mid latitude westerly jet and a wave function representing an ocean land topography they truncated the equation into a three spectral mode model which gives rise to the existence of multiple equilibrium states showing a blocking mechanism profile moreover they predicted numerically the absence of the multiple equilibrium states with respect to the flat topography situation in the present paper this absence observation is discussed from rigorous analysis and the coexistence of two stable equilibrium states similarly to the three mode model equilibrium states accounting for atmospheric blocking is investigated numerically with the increment of topographic amplitude

stability and bifurcation of the taylor problem the main objective of this paper is to address the stability and bifurcation of the couette flow between two concentric rotating cylinders and to verify rigorously taylor s observation in his experiments 13 a nonlinear theory is obtained for the taylor problem leading in particular to rigorous justifications of the linear theory used by physicists and the taylor vortex structure the main technical tools are the dynamic bifurcation theory and the geometric theory for incompressible flows both developed recently by ma wang 10 12

inequivalent complexity criteria for free boundary minimal surfaces abstract we obtain a series of results in the global theory of free boundary minimal surfaces which in particular provide a rather complete picture for the way different complexity criteria such as area topology and morse index compare beyond the regime where effective estimates are at disposal

hopf bifurcation in quasi geostrophic channel flow in this article we conduct a rigorous stability and bifurcation analysis for a highly idealized model of planetary scale atmospheric and oceanic flows the model is governed by the two dimensional quasi geostrophic equation for the conservation of vorticity in an east west oriented periodic channel the main result is the existence of hopf bifurcation of the flow as the reynolds number crosses a critical value the key idea in proving this result is translating the eigenvalue problem into a difference equation and treating the latter by continued fraction methods numerical results are obtained by using a finite difference scheme with high spatial resolution and these results agree closely with the theoretical predictions the spatio temporal structure of the limit cycle corresponds to a wave that propagates slowly westward and is symmetric about the midaxis of the channel for plausible paramater values that correspond to midlatitude atmospheric flows the period of this wave is 20 25 days

nonmatrix varieties and nil generated algebras whose units satisfy a group identity abstract letr denote the group of units of an associative algebrarover an infinite fieldf we prove that ifris unitarily generated by its nilpotent elements thenr satisfies a group identity precisely whenrsatisfies a nonmatrix polynomial identity as an application we examine the group algebrafgof a torsion groupgand the restricted enveloping algebrau l of ap nil restricted lie algebral giambruno sehgal and valenti recently proved that if the group of units fg satisfies a group identity thenfgsatisfies a polynomial identity thus confirming a conjecture of brian hartley we show that in fact fg satisfies a group identity if and only iffgsatisfies a nonmatrix polynomial identity in the case of restricted enveloping algebras we prove thatu l satisfies a group identity if and only ifu l satisfies the engel condition

vanishing viscosity limit for global attractors for the damped navier stokes system with stress free boundary conditions abstract we consider the damped and driven navier stokes system with stress free boundary conditions and the damped euler system in a bounded domain r 2 we show that the damped euler system has a strong global attractor in h 1 we also show that in the vanishing viscosity limit the global attractors of the navier stokes system converge in the non symmetric hausdorff distance in h 1 to the strong global attractor of the limiting damped euler system whose solutions are not necessarily unique

a dichotomy result for prime algebras of gelfand kirillov dimension two abstract let k be an uncountable field we show that a finitely generated prime goldie k algebra of quadratic growth is either primitive or satisfies a polynomial identity answering a question of small in the affirmative

the isoperimetric problem on surfaces of revolution of decreasing gauss curvature we prove that the least perimeter way to enclose prescribed area in the plane with smooth rotationally symmetric complete metric of nonincreasing gauss curvature consists of one or two circles bounding a disc the complement of a disc or an annulus we also provide a new isoperimetric inequality in general surfaces with boundary

rational subgroups of biautomatic groups centralizers of finite subsets in biautomatic groups are them selves biautomatic every polycyclic subgroup of a biautomatic group is abelian by finite

prescribing curvature on open surfaces in this article we study the problem sometimes called the berger nirenberg problem of prescribing the curvature on a riemann surface that is on an oriented surface equipped with a conformal class of riemannian metrics

steady state bifurcation analysis of a strong nonlinear atmospheric vorticity equation the quasi geostrophic vorticity equation studied in the present paper is a simplified form of the atmospheric circulation model introduced by charney and devore j atmos sci 36 1979 1205 1216 on the existence of multiple steady states to the understanding of the persistence of atmospheric blocking the fluid motion defined by the equation is driven by a zonal thermal forcing and an ekman friction forcing measured by it is proved that the steady state solution is globally unique for large values while multiple steady state solutions branch off the basic steady state solution for crit where the critical value crit is less than one without involvement of viscosity the equation has fully non linear property as its non linear part contains the highest order derivative term steady state bifurcation analysis is essentially based on the compactness which can be simply obtained for semilinear equations such as the navier stokes equations but is not available for the fully nonlinear quasi geostrophic vorticity equation in the euler formulation therefore the lagrangian formulation of the equation is employed to gain the required compactness

quantitative rigidity results for conformal immersions in this paper we prove several quantitative rigidity results for conformal immersions of surfaces in with bounded total curvature we show that branched conformal immersions which are close in energy to either a round sphere a conformal clifford torus an inverted catenoid an inverted enneper s minimal surface or an inverted chen s minimal graph must be close to these surfaces in the w norm moreover we apply these results to prove a corresponding rigidity result for complete connected and non compact surfaces

computational design for long term numerical integration of the equations of fluid motion the integral constraints on quadratic quantities of physical importance such as conservation of mean kinetic energy and mean square vorticity will not be maintained in finite difference analogues of the equation of motion for two dimensional incompressible flow unless the finite difference jacobian expression for the advection term is restricted to a form which properly represents the interaction between grid points as derived in this paper it is shown that the derived form of the finite difference jacobian prevents nonlinear computational instability and thereby permits long term numerical integrations

hopf bifurcation and transition of three dimensional wind driven ocean circulation problem abstract the purpose of this article is three fold the first is to investigate the linear stability of the basic flow governed by the three dimensional 3d quasi geostrophic qg equation with a steady wind force we show that the basic flow will become unstable as the reynolds number crosses a critical value the second is to study the dynamic transition associated with the instability of the basic flow at the critical reynolds number it is proved that the type of the phase transition is determined by a parameter called transition number which is mainly determined by the reynolds number and the aspect ratio more precisely the negative positive sign of the real part of the transition number corresponds to the continuous catastrophic transition the last objective is to apply numerical experiments to examine the specific transition type for specified control parameters we show that the real part of the transition number is always negative leading to only the continuous transition

small cancellations over relatively hyperbolic groups and embedding theorems we generalize the small cancellation theory over ordinary hyperbolic groups to relatively hyperbolic settings this generalization is then used to prove various embedding theorems for countable groups for instance we show that any countable torsion free group can be embedded into a finitely generated group with exactly two conjugacy classes in particular this gives the affirmative answer to the well known question of the existence of a finitely generated group g other than 2 such that all nontrivial elements of g are conjugate

the nilpotency of the radical in a finitely generated p i ring abstract let r x 1 x k be a p i ring satisfying a monic polynomial identity one of its coefficients is 1 where is a central noetherian subring it is proven that n r the nil radical of r is nilpotent as a corollary by taking f a field we settle affirmatively the open problem posed in c procesi rings with polynomial identities p 186 marcel dekker new york 1973 we prove the jacobson radical of a finitely generated p i algebra is nilpotent

a two level finite element discretization of the streamfunction formulation of the stationary quasi geostrophic equations of the ocean in this paper we proposed a two level finite element discretization of the nonlinear stationary quasi geostrophic equations which model the wind driven large scale ocean circulation optimal error estimates for the two level finite element discretization were derived numerical experiments for the two level algorithm with the argyris finite element were also carried out the numerical results verified the theoretical error estimates and showed that for the appropriate scaling between the coarse and fine mesh sizes the two level algorithm significantly decreases the computational time of the standard one level algorithm

topology of negatively curved real affine algebraic surfaces we find a quartic example of a smooth embedded negatively curved surface in r homeomorphic to a doubly punctured torus this constitutes an explicit solution to hadamard s problem on constructing complete surfaces with negative curvature and euler characteristic in r further we show that our solution has the optimal degree of algebraic complexity via a topological classification for smooth cubic surfaces with a negatively curved component in r any such component must either be topologically a plane or an annulus in particular we prove that there exists no cubic solutions to hadamard s problem

on the bass lubotzky question about quotients of hyperbolic groups abstract we prove at theorem 1 that any non elementary hyperbolic group g possesses a non trivial finitely presented quotient q having no non trivial subgroups of finite indices the theorem was commissioned in august 1997 by h bass and a lubotzky because the statement was required to constructing of their counter examples to platonov s conjecture on the representation rigid linear groups bl theorem 2 gives three equivalent reformulations of the known problem on existence of non residually finite hyperbolic groups

l2 stability for solutions of subcritical quasi geostrophic equation with large perturbations abstract this paper concerns with the asymptotic stability for the weak solution of the two dimensional dissipative subcritical quasi geostrophic equation in the class l p 0 t l q r 2 for p 2 q 2 q 1 2 it is proved that although the initial data and external force perturbations from are large every perturbed weak solution converges asymptotically to in l 2 r 2 as t as a byproduct we also obtain the weak strong uniqueness criterion of subcritical quasi geostrophic equation which improves the earlier results

total mean curvature of positively curved hypersurfaces abstract in this article we prove that every positively curved complete non compact hypersurface in r n has infinite total mean curvature

on the polynomial representations of current algebras abstract we obtain a sufficient condition for a representation of a lie algebra of the form l g where g is a finite dimensional simple lie algebra over an algebraically closed field f of zero characteristic and is an associative commutative algebra over f to be polynomial as a corollary we classify integrable graded irreducible modules over an extended loop algebra with finite dimensional homogeneous spaces the criteria is applied also to irreducible conformal modules over current algebras

the barotropic vorticity equation under forcing and dissipation bifurcations of nonsymmetric responses and multiplicity of solutions a simplified model of barotropic flow in a bounded domain has been used and stationary responses due to the balance between dissipation and external forcing have been studied the forcing as well as the domain is assumed to admit some symmetry group sufficient conditions are deduced by which a branch of nonsymmetric stationary solutions bifurcate from a symmetric one and those results are applied to a specific problem of zonally symmetric forcing in an open channel

global well posedness and finite dimensional global attractor for a 3 d planetary geostrophic viscous model in this paper we consider a three dimensional planetary geostrophic viscous model of the gyre scale mid latitude ocean we show the global existence and uniqueness of the weak and strong solutions to this model moreover we establish the existence of a nite dimensional global attractor to this dissipative evolution system c 2003 wiley periodicals inc

cmc biconservative surfaces in sn r and hn r abstract we classify non minimal biconservative surfaces with parallel mean curvature vector field in s n r and h n r when these surfaces do not lie in s n or h n and they are not vertical cylinders we find their explicit local equation we also prove a result on the compactness of biconservative surfaces with constant mean curvature in hadamard manifolds

volume growth and bernstein theorems for translating solitons abstract in this paper we study properties of the potential function of a translating soliton m in r n 1 and the volume growth of the intersection of euclidean balls with m we show that if m is a convex translating soliton then the infimum of the mean curvature function on m is zero i e inf m h 0 we give a condition to obtain the bernstein theorem for the translating solitons we also give an outline of a simple proof of the bernstein theorem due to bombieri de giorgi miranda

nonpositively curved surfaces in r 3 we consider complete nonpositively curved surfaces with one end twice continuously differentiably immersed in euclidean three space if such a surface is embedded near infinity and has square integrable second fundamental form then it must lie between two parallel planes we generalize some results obtainedby h chan 2 3 about asymptotically totally geodesic complete nonpositively curved surfaces c 2 immersedin r 3 we show that if such a surface has one endwhich is embedded near infinity and satisfies the condition that the second fundamental form is square integrable then it must be contained in a slab the closedregion between two parallel planes moreover the surface must contact the bounding planes of the narrowest slab along locally nonconvex sets heading to infinity this implies h chan s result that a one ended surface with nonpositive curvature having only isolatedparabolic points ad mits no isometric immersion of this type at all 2 3 an example of such a surface is the solution of the equation

waves vs particles in the atmosphere s phase space a pathway to long range forecasting thirty years ago e n lorenz provided some approximate limits to atmospheric predictability the details in space and time of atmospheric flow fields are lost after about 10 days certain gross flow features recur however after times of the order of 10 50 days giving hope for their prediction over the last two decades numerous attempts have been made to predict these recurrent features the attempts have involved on the one hand systematic improvements in numerical weather prediction by increasing the spatial resolution and physical faithfulness in the detailed models used for this prediction on the other hand theoretical attempts motivated by the same goal have involved the study of the large scale atmospheric motions9 phase space and the inhomogeneities therein these coarse graining studies have addressed observed as well as simulated atmospheric data sets two distinct approaches have been used in these studies the episodic or intermittent and the oscillatory or periodic the intermittency approach describes multiple flow or weather regimes their persistence and recurrence and the markov chain of transitions among them the periodicity approach studies intraseasonal oscillations with periods of 15 70 days and their predictability we review these two approaches particles vs waves in the quantum physics analogy alluded to in the title of this article discuss their complementarity and outline unsolved problems

radical rings with engel conditions abstract an associative ring r without unity is called radical if it coincides with its jacobson radical which means that the set of all elements of r forms a group denoted by r under the circle operation r s r s rs on r it is proved that for a radical ring r the group r satisfies an n engel condition for some positive integer n if and only if r is m engel as a lie ring for some positive integer m depending only on n

global well posedness of the three dimensional viscous primitive equations of large scale ocean and atmosphere dynamics in this paper we prove the global existence and uniqueness regularity of strong solutions to the three dimensional viscous primitive equations which model large scale ocean and atmosphere dynamics 1 introduction large scale dynamics of oceans and atmosphere is governed by the primitive equations which are derived from the navier stokes equations with rotation coupled to thermodynamics and salinity diffusion transport equations which account for the buoyancy forces and stratification effects under the boussinesq approximation moreover and due to the shallowness of the oceans and the atmosphere i e the depth of the fluid layer is very small in comparison to the radius of the earth the vertical large scale motion in the oceans and the atmosphere is much smaller than the horizontal one which in turn leads to modeling the vertical motion by the hydrostatic balance as a result

pi algebras generated by nilpotent elements of bounded index abstract let r be an associative algebra over a field f of positive characteristic p we address the following problem if r is generated by nilpotent elements with bounded index under what conditions can we conclude that r itself is nil of bounded index we prove that whenever r satisfies the engel condition and p 0 or r is lie soluble and p 2 then the conclusion holds along the way we prove that if r satisfies the engel condition then there exists a positive integer n such that the map x x p n is additive the converse also holds when f is infinite or r is generated by nilpotent elements

structure of prime finitely presented monomial algebras abstract the structure of a finitely presented monomial algebra k x k i over a field k is described here x is a finitely generated free monoid and i is a prime ideal of x that is finitely generated as an application a new structural proof of the recent result of bell and pekcagliyan j bell p pekcagliyan primitivity of finitely presented monomial algebras preprint arxiv 0712 0815v1 on the primitivity of such algebras is presented which yields a positive solution to the trichotomy problem raised by bell and smoktunowicz j bell a smoktunowicz the prime spectrum of algebras of quadratic growth j algebra 319 2008 414 431 in the finitely presented case our approach is based on a new result on the form of prime rees factors of semigroups satisfying the ascending chain condition on one sided annihilators and on its refinement in the case of finitely presented factors of the form x i

when is a power of the frobenius map on a noncommutative ring a homomorphism abstract let x x p be the frobenius map on an associative unital ring r with prime characteristic p it is well known that whenever r is commutative n is a ring homomorphism for all positive integers n the converse however is not true in general indeed we prove that if r is m engel for some positive integer m then there exists a positive integer n 0 depending only on m such that for all n n 0 n is a ring homomorphism with central image conversely if any one of the following conditions holds n respects addition n respects multiplication n respects lie multiplication or the image of n is commutative then r is m engel for some m depending only on n consequently if is surjective and any one of the aforementioned conditions holds then r must be commutative

tangent cones and regularity of real hypersurfaces we characterize c embedded hypersurfaces of r as the only locally closed sets with continuously varying flat tangent cones whose measuretheoretic multiplicity is at most m 3 2 it follows then that any topological hypersurface which has flat tangent cones and is supported everywhere by balls of uniform radius is c in the real analytic case the same conclusion holds under the weakened hypothesis that each tangent cone be a hypersurface in particular any convex real analytic hypersurface x r is c furthermore if x is real algebraic strictly convex and unbounded then its projective closure is a c hypersurface as well which shows that x is the graph of a function defined over an entire hyperplane