the as cohen macaulay property for quantum flag manifolds of minuscule weight abstract it is shown that quantum homogeneous coordinate rings of generalised flag manifolds corresponding to minuscule weights their schubert varieties big cells and determinantal varieties are as cohen macaulay the main ingredient in the proof is the notion of a quantum graded algebra with a straightening law introduced by t h lenagan and l rigal t h lenagan l rigal quantum graded algebras with a straightening law and the as cohen macaulay property for quantum determinantal rings and quantum grassmannians j algebra 301 2006 670 702 using stanley s theorem it is moreover shown that quantum generalised flag manifolds of minuscule weight and their big cells are as gorenstein

the cyclicity of period annuli of a class of quintic hamiltonian systems abstract in this paper we prove that the cyclicity of period annuli for system d x d t y d y d t a x b x 3 c x 5 under perturbations of polynomials with degree n is not more than 54 n 13 taking into account the multiplicity where a b c r with c 0 such that the unperturbed systems have at least one center

quantum analogues of richardson varieties in the grassmannian and their toric degeneration abstract in the present paper we are interested in natural quantum analogues of richardson varieties in the type a grassmannians to be more precise the objects that we investigate are quantum analogues of the homogeneous coordinate rings of richardson varieties which appear naturally in the theory of quantum groups our point of view here is geometric we are interested in the regularity properties of these non commutative varieties such as their irreducibility normality cohen macaulayness in the spirit of non commutative algebraic geometry a major step in our approach is to show that these algebras have the structure of an algebra with a straightening law from this it follows that they degenerate to some quantum analogues of toric varieties

the equation for time like extremal surfaces in minkowski space r2 n in this paper we investigate the equation for time like extremal surfaces in the minkowski space r2 n and show that this kind of equation enjoys many interesting properties nonstrict hyperbolicity constant multiplicity of eigenvalues boundedness of characteristic propagation speeds linear degeneracy of all characteristic fields richness etc without any smallness assumption of initial data we give the necessary and sufficient condition on the global existence of classical solutions of the cauchy problem based on this we prove some global existence theorems on classical solutions of the dirichlet problem and neumann problem for this kind of equation finally we present an explicit exact representation involving two independent arbitrary functions of general solution

geometry of quasilinear hyperbolic systems with characteristic fields of constant multiplicity in this paper we study the regularity of the eigenvalues and eigenvectors and the existence of normalized coordinates for quasilinear hyperbolic systems with characteristic fields of constant multiplicity we prove that the eigenvalues and eigenvectors of the system have the same regularity as the coefficients of the system on the other hand we show that for the quasilinear hyperbolic system of conservation laws with characteristic fields of constant multiplicity the normalized coordinates exist on the domain under consideration

almost global existence of classical discontinuous solutions to genuinely nonlinear hyperbolic systems of conservation laws with small bv initial data abstract in the present paper the author investigates the initial problem for genuinely nonlinear quasilinear hyperbolic systems of conservation laws under small bv perturbations of the riemann initial data where the perturbations are in bv but they are assumed to be c 1 smooth with bounded and possibly large c 1 norms combining the techniques employed by li and kong with the modified glimm s functional the author obtains the almost global existence and lifespan of classical discontinuous solutions to the above problem this result is also applied to the system of one dimensional isentropic flow

the generalized nonlinear initial boundary riemann problem for linearly degenerate quasilinear hyperbolic systems of conservation laws abstract this work is a continuation of our previous work in the present paper we study the generalized nonlinear initial boundary riemann problem with small bv data for linearly degenerate quasilinear hyperbolic systems of conservation laws with nonlinear boundary conditions in a half space t x t 0 x 0 we prove the global existence and uniqueness of piecewise c 1 solution containing only contact discontinuities to a class of the generalized nonlinear initial boundary riemann problem which can be regarded as a small bv perturbation of the corresponding nonlinear initial boundary riemann problem for general n n linearly degenerate quasilinear hyperbolic system of conservation laws moreover this solution has a global structure similar to the one of the self similar solution u u x t to the corresponding nonlinear initial boundary riemann problem some applications to quasilinear hyperbolic systems of conservation laws arising in the string theory and high energy physics are also given

primitive ideals of cq sl n abstract the primitive ideals of the quantum group c q sl n are classified in the case where q is a non zero complex number which is not a root of unity it is shown that the orbits in prim c q sl n under the action of the character group h c n 1 are parameterized naturally by w w where w is the associated weyl group it is shown that there is a natural one to one correspondence between primitive ideals of c q sl n and symplectic leaves of the associated poisson algebraic group sl n c

perturbation from an elliptic hamiltonian of degree four iii global centre abstract the paper deals with lienard equations of the form x y y p x yq x with p and q polynomials of degree respectively 3 and 2 attention goes to perturbations of the hamiltonian vector fields with an elliptic hamiltonian of degree four exhibiting a global centre it is proven that the least upper bound of the number of zeros of the related elliptic integral is four and this is a sharp one this result permits to prove the existence of lienard equations of type 3 2 with a quadruple limit cycle with both a triple and a simple limit cycle with two semistable limit cycles with one semistable and two simple limit cycles or with four simple limit cycles

the mixed initial boundary value problem for diagonalizable quasilinear hyperbolic systems in the first quadrant abstract in this paper we investigate the mixed initial boundary value problem for diagonalizable quasilinear hyperbolic systems with nonlinear boundary conditions on a half unbounded domain t x t 0 x 0 under the assumptions that system is strictly hyperbolic and linearly degenerate we obtain the global existence and uniqueness of c 1 solutions with the bounded l 1 l norm of the initial data as well as their derivatives and appropriate boundary condition based on the existence results of global classical solutions we also prove that when t tends to infinity the solutions approach a combination of c 1 travelling wave solutions under the appropriate assumptions of initial and boundary data the results can be applied to the equation of time like extremal surface in minkowski space r 1 1 n

prime spectra of quantum semisimple groups we study the prime ideal spaces of the quantized function algebras rq g for g a semisimple lie group and q an indeterminate our method is to examine the structure of algebras satisfying a set of seven hypotheses and then to demonstrate using work of joseph hodges and levasseur that the algebras rq g satisfy this list of assumptions rings satisfying the assumptions are shown to satisfy normal separation and therefore jategaonkar s strong second layer condition for such rings much representation theoretic information is carried by the graph of links of the prime spectrum and so we proceed to a detailed study of the prime links of algebras satisfying the list of assumptions homogeneity is a key feature it is proved that the clique of any prime ideal coincides with its orbit under a finite rank free abelian group of automorphisms bounds on the ranks of these groups are obtained in the case of rq g in the final section the results are specialized to the case g sln c where detailed calculations can be used to illustrate the general results as a preliminary set of examples we show also that the multiparameter quantum coordinate rings of affine n space satisfy our axiom scheme when the group generated by the parameters is torsionfree 0 introduction and background 0 1 let p and q be prime ideals of a noetherian ring r we say that p is linked to q and write p q if there is an ideal a of r with pq c a c p n q such that p n q a is a nonzero torsionfree left r p module and a torsionfree right r q module this condition can be simplified in case r satisfies the second layer condition see below in that case p q if and only if p n q pq is faithful as a left r p module and as a right r q module 39 10 1 4 the graph of links of r is the directed graph whose vertices are the points of spec r the set of prime ideals of r with a directed edge from p to q if and only if p q the primes in the connected component of the graph of links containing p constitute the clique of p denoted clique p there is a close connection between the graph of links of r and the representation theory of r cf 20 4 this connection is particularly strong when r satisfies the strong second layer condition which requires that whenever u is a cyclic uniform r module with the annihilator received by the editors november 4 1994 and in revised form september 5 1995 1991 mathematics subject classification primary 16d30 16d60 16p40 17b37 the research of the second author was partially supported by a grant from the national science foundation usa part of the work was carried out while he visited the university of glasgow mathematics department during october 1993 supported by the edinburgh and london mathematical societies work on a revised version of the paper was completed in summer 1995 during a visit by both authors to the department of mathematics of the university of washington whom both thank for its hospitality the travel costs of the first author were in part covered by a grant from the carnegie trust for the universities of scotland 1996 american mathematical society

perturbations from an elliptic hamiltonian of degree four abstract the paper deals with lienard equations of the form x y y p x yq x with p and q polynomials of degree respectively 3 and 2 attention goes to perturbations of the hamiltonian vector field with an elliptic hamiltonian of degree 4 exhibiting a cuspidal loop it is proven that the least upper bound for the number of zeros of the related elliptic integral is four and this upper bound is a sharp one this permits to prove the existence of lienard equations of type 3 2 with at least four limit cycles the paper also contains a complete result on the respective number of small and large limit cycles

connected graded gorenstein algebras with enough normal elements we generalize 12 1 1 and 1 2 to the following situation theorem 1 let a be a connected graded noetherian algebra of injective dimension d such that every nonsimple graded prime factor ring of a contains a homogeneous normal element of positive degree then 1 a is auslander gorenstein and cohen macaulay 2 a has a quasi frobenius quotient ring 3 every minimal prime ideal p is graded andgkdim a p d 4 if moreover a has finite global dimension then a is a domain and a maximal order in its quotient division ring to prove the above we need the following result which is a generalization of 3 2 46 ii theorem 2 let a be a connected graded noetherian as gorenstein algebra of injective dimension d then 1 the last term of the minimal injective resolution of aais isomorphic to a shift of a 2 for every noetherian graded a module m extd m a is finite dimensional over k

lifespan of classical discontinuous solutions to general quasilinear hyperbolic systems of conservation laws with small bv initial data rarefaction waves abstract in the present paper the author investigates the global structure stability of riemann solutions for general quasilinear hyperbolic systems of conservation laws under small bv perturbations of the initial data where the riemann solution contains rarefaction waves while the perturbations are in bv but they are assumed to be c 1 smooth with bounded and possibly large c 1 norms combining the techniques employed by li kong with the modified glimm s functional the author obtains a lower bound of the lifespan of the piecewise c 1 solution to a class of generalized riemann problems which can be regarded as a small bv perturbation of the corresponding riemann problem this result is also applied to the system of traffic flow on a road network using the aw rascle model

a class of quadratic matrix algebras arising from the quantized enveloping algebra uq a2n 1 a natural family of quantized matrix algebras is introduced it includes the two best studied such located inside uq a2n 1 it consists of quadratic algebras with the same hilbert series as polynomials in n2 variables we discuss their general properties and investigate some members of the family in great detail with respect to associated varieties degrees centers and symplectic leaves finally the space of rank r matrices becomes a poisson submanifold and there is an associated tensor category of rank r matrices

blow up of periodic solutions to linearly degenerate quasilinear hyperbolic systems an example is given to show that for the cauchy problem of n n homogeneous linearly degenerate quasilinear hyperbolic systems with periodic initial data when n 2 c1 solutions may blow up in a finite time no matter how small and smooth the initial data are

global existence of classical solutions to the mixed initial boundary value problem for quasilinear hyperbolic systems of diagonal form with large bv data abstract in this paper we investigate the mixed initial boundary value problem with large bv data for linearly degenerate quasilinear hyperbolic systems of diagonal form with general nonlinear boundary conditions in the half space t x t 0 x 0 as the result in a bressan contractive metrics for nonlinear hyperbolic systems indiana univ math j 37 1988 409 421 suggests that one may achieve global smoothness even if the c 1 norm of the initial data is large we prove that if the c 1 norm and the bv norm of the initial and boundary data are bounded but possibly large then the solution remains c 1 globally in time and possesses uniformly bounded total variation in x for all t 0 as an application we apply the result to the system describing the motion of relativistic closed strings in the minkowski space r 1 n

the infinitesimal 16th hilbert problem in the quadratic case let h x y be a real cubic polynomial with four distinct critical values in a complex domain and let x h h y frac partial partial x h x frac partial partial y be the corresponding hamiltonian vector field we show that there is a neighborhood of x h in the space of all quadratic plane vector fields such that any x has at most two limit cycles

cyclicity of periodic annulus and hopf cyclicity in perturbing a hyper elliptic hamiltonian system with a degenerate heteroclinic loop abstract in this paper we study the cyclicity of periodic annulus and hopf cyclicity in perturbing a quintic hamiltonian system the undamped system is hyper elliptic non symmetric with a degenerate heteroclinic loop which connects a hyperbolic saddle to a nilpotent saddle we rigorously prove that the cyclicity is 3 for periodic annulus when the weak damping term has the same degree as that of the associated hamiltonian system this result provides a positive answer to the open question whether the annulus cyclicity is 3 or 4 when the smooth polynomial damping term has degree n first a transformation based on the involution of the hamiltonian is introduced and then we analyze the coefficients involved in the bifurcation function to show that the hopf cyclicity is 2 n 1 3 further for piecewise smooth polynomial damping with a switching manifold at the y axis we consider the damping terms to have degrees l and n respectively and prove that the hopf cyclicity of the origin is 3 l 2 n 4 3 3 n 2 l 4 3 when l n n l

hochschild and cyclic homology of quantum groups for an arbitrary complex linear semisimple lie groupg we consider hopf algebras of the deformations of the formal and algebraic functions ong the hochschild and cyclic homology of these hopf algebras are computed when the value of the deformation parameter is generic

asymptotic behavior of classical solutions of reducible quasilinear hyperbolic systems with characteristic boundaries abstract in this paper we investigate the asymptotic behavior of classical solutions of reducible quasilinear hyperbolic systems with characteristic boundaries under some suitable assumptions we prove that the solution approaches a combination of lipschitz continuous and piecewise c 1 traveling wave solution as an application we apply the result to the equation for time like extremal surfaces in the minkowski space time r 1 1 n

lifespan of classical discontinuous solutions to general quasilinear hyperbolic systems of conservation laws with small bv initial data shocks and contact discontinuities abstract this work is a continuation of our previous work z q shao global structure stability of riemann solutions for linearly degenerate hyperbolic conservation laws under small bv perturbations of the initial data nonlinear anal real world appl 11 2010 3791 3808 in the present paper we investigate the global structure stability of riemann solutions for general quasilinear hyperbolic systems of conservation laws under small bv perturbations of the initial data where the riemann solution only contains the shocks and contact discontinuities and at least a shock wave the perturbations are in bv but they are assumed to be c 1 smooth with bounded and possibly large c 1 norms we get a lower bound of the lifespan of the piecewise c 1 solution to a class of the generalized riemann problem which can be regarded as a small bv perturbation of the corresponding riemann problem some applications to quasilinear hyperbolic systems of conservation laws arising in physics particularly to one dimensional euler equations of gas dynamics for a compressible inviscid non heat conducting gas in eulerian coordinates are also given

completely linear degeneracy for quasilinear hyperbolic systems abstract in this paper we introduce a new concept of completely linear degeneracy for quasilinear hyperbolic systems in several space variables and then get an interesting property for multidimensional hyperbolic conservation laws satisfying our new definition for applications we give some examples arising from mathematics and physics at last

rings with auslander dualizing complexes abstract a ring with an auslander dualizing complex is a generalization of an auslander gorenstein ring we show that many results which hold for auslander gorenstein rings also hold in the more general setting on the other hand we give criteria for existence of auslander dualizing complexes which show these occur quite frequently the most powerful tool we use is the local duality theorem for connected graded algebras over a field filtrations allow the transfer of results to nongraded algebras we also prove some results of a categorical nature most notably the functoriality of rigid dualizing complexes

perturbation from an elliptic hamiltonian of degree four iv figure eight loop the paper deals with lienard equations of the form x y y p x yq x with p and q polynomials of degree respectively 3 and 2 attention goes to perturbations of the hamiltonian vector fields with an elliptic hamiltonian of degree four exhibiting a figure eight loop it is proved that the least upper bound of the number of zeros of the related elliptic integral is five and this is a sharp bound multiplicity taken into account moreover if restricting to the level curves inside a saddle loop or outside the figure eight loop the sharp upper bound is respectively two or four also the multiplicity of the zeros is at most four this is the last one in a series of papers on this subject the results of this paper together with j differential equations 176 2001 114 j differential equations 175 2001 209 j differential equations to be published largely finish the study of the cubic perturbations of the elliptic hamiltonians of degree four and presumably provide a complete description of the number and the possible configurations of limit cycles for cubic lienard equations with small quadratic damping as a special case we obtain a configuration of four limit cycles surrounding three singularities together with a small limit cycle which surrounds one of the singularities

perturbations of a class of hyper elliptic hamiltonian systems of degree seven with nilpotent singular points in this paper we study the number of isolated zeros of abelian integrals associated to system x y y x 3 x 2 1 2 under the perturbations of 0 1 x 2 2 x 4 3 x 6 y y where 0 1 and i r i 0 1 2 3 we find a special transformation such that four generating elements are reduced to special three ones and then the algebraic method can be used the bounds and sharp bounds are obtained

quantum deformations of certain simple modules over enveloping algebras soit f un corps soit f 0 un sous corps de f soit u l algebre enveloppante universelle sur f 0 de l algebre de lie de kac moody correspondant a la matrice a d entiers on montre qu un module integrable de poids le plus eleve v de u admet une deformation quantique

bifurcations of limit cycles from quintic hamiltonian systems with an eye figure loop abstract in this paper we consider lienard equations of the form x y y x 2 x 3 x 5 e x 2 x 4 y where 0 e 1 r 3 and is bounded we prove that the least upper bound for the number of zeros of the related abelian integrals i h h x 2 x 4 y d x is 2 taking into account their multiplicities for h 0 1 6 and this upper bound is a sharp one this implies that the number of limit cycles bifurcated from periodic orbits in the vicinity of the center of the unperturbed system for e 0 inside an eye figure loop is less than or equal to 2

bifurcations of limit cycles for a quintic hamiltonian system with a double cuspidal loop in this work we consider the number of limit cycles that can bifurcate from periodic orbits located inside a double cuspidal loop of the quintic hamiltonian vector field x h y x x 3 x 2 1 y under small perturbations of the form e a bx 2 cx 4 y y where 0 e 1 and a b c are real constants using picard fuchs equations for related abelian integrals asymptotic expansion of these integrals about critical level curves of h and some geometric properties of the curves defined by ratios of two especial integrals we show that the least upper bound for the number of limit cycles appeared in this bifurcation is two

global existence and asymptotic behavior of classical solutions of quasilinear hyperbolic systems with linearly degenerate characteristic fields abstract in this paper we study the global existence and the asymptotic behavior of classical solution of the cauchy problem for quasilinear hyperbolic system with constant multiple and linearly degenerate characteristic fields we prove that the global c 1 solution exists uniquely if the bv norm of the initial data is sufficiently small based on the existence result on the global classical solution we show that when the time t tends to the infinity the solution approaches a combination of c 1 traveling wave solutions finally we give an application to the equation for time like extremal surfaces in the minkowski space time r 1 n

exact bound on the number of zeros of abelian integrals for two hyper elliptic hamiltonian systems of degree 4 abstract this paper concerns the exact bound on the number of zeros of abelian integrals associated with two hyper elliptic hamiltonian systems of degree 4 the upper and lower bounds for the two systems have been obtained in several previous works however the sharp bounds are still unknown in this paper we provide a proof to show that the exact bound is 3 for both systems the basic idea of our method is to bound the parameter space in r 3 to obtain two parameter sets which might yield maximal 4 zeros of abelian integrals corresponding to the two sets further we show that the existence of 4 zeros on the two sets is not possible and thus the sharp bound is 3

linear estimate of the number of zeros of abelian integrals for a kind of quartic hamiltonians abstract an upper bound b n 7n 5 is derived for the number of zeros of abelian integrals i h h g x y dy f x y dx on the open interval where h is an oval lying on the algebraic curve h x y 1 2 y2 u x h deg u x 4 and is the maximal interval of existence of h f x y g x y are polynomials of x and y and n max deg f x y deg g x y

winding invariant prime ideals in quantum 3 3 matrices abstract a complete determination of the prime ideals invariant under winding automorphisms in the generic 3 3 quantum matrix algebra o q m 3 k is obtained explicit generating sets consisting of quantum minors are given for all of these primes thus verifying a general conjecture in the 3 3 case the result relies heavily on certain tensor product decompositions for winding invariant prime ideals developed in an accompanying paper in addition new methods are developed here which show that certain sets of quantum minors not previously manageable generate prime ideals in o q m n k

bifurcations of limit cycles from quintic hamiltonian systems with an eye figure loop ii abstract in this paper we consider lienard equations of the form x y y x 2 x 3 x 5 e x 2 x 4 y where 0 e 1 r 3 and is bounded we prove that the least upper bound for the number of zeros of the related abelian integrals i h h x 2 x 4 y d x for h 1 6 is three and for h 0 is four counted with multiplicity for all parameters and this implies that the number of limit cycles that bifurcated from periodic orbits of the unperturbed system for e 0 outside an eye figure loop is less than or equal to three

semilinear behavior for totally linearly degenerate hyperbolic systems with relaxation we investigate totally linearly degenerate hyperbolic systems with relaxation we aim to study their semilinear behavior which means that the local smooth solutions cannot develop shocks and the global existence is controlled by the supremum bound of the solution in this paper we study two specific examples the it suliciu type and the it kerr debye type models for the suliciu model which arises from the numerical approximation of isentropic flows the semilinear behavior is obtained using pointwise estimates of the gradient for the kerr debye systems which arise in nonlinear optics we show the semilinear behavior via energy methods for the original kerr debye model thanks to the special form of the interaction terms we can show the global existence of smooth solutions

global existence and asymptotic behavior of solutions to a nonlinear wave equation of fourth order in this paper we focus on the cauchy problem for a nonlinear wave equation of fourth order in n dimensional space n 1 the decay structure of which is of regularity loss property based on the decay estimate of solutions to the linear problem we introduce a set of time weighted sobolev spaces by using the contraction mapping theorem we obtain the global in time existence and the optimal decay estimates of solutions to the cauchy problem under smallness assumption on the initial data

global existence and asymptotic behavior of solutions for a semi linear wave equation abstract in this paper we investigate the initial value problem for a semi linear wave equation in n dimensional space based on the decay estimate of solutions to the corresponding linear equation we define a set of time weighted sobolev spaces under small condition on the initial value we prove the global existence and asymptotic behavior of the solution in the corresponding sobolev spaces by the contraction mapping principle

transfer results for frobenius extensions abstract we study frobenius extensions which are free filtered by a totally ordered finitely generated abelian group and their free graded counterparts first we show that the frobenius property passes up from a free graded extension to a free filtered extension then also from a free filtered extension to the extension of their rees algebras our main theorem states that under some natural hypotheses a free filtered extension of algebras is frobenius if and only if the associated graded extension is frobenius in the final section we apply this theorem to provide new examples and non examples of frobenius extensions

the mixed initial boundary value problem for quasilinear hyperbolic systems with linearly degenerate characteristics abstract in this paper we investigate the mixed initial boundary value problem with small bv data for linearly degenerate quasilinear hyperbolic systems with general nonlinear boundary conditions in the half space t x t 0 x 0 as the result in a bressan contractive metrics for nonlinear hyperbolic systems indiana univ math j 37 1988 409 421 suggests that one may achieve global smoothness even if the c 1 norm of the initial data is large we prove that if the c 1 norm of the initial and boundary data is bounded but possibly large and the bv norm of the initial and boundary data is sufficiently small then the solution remains c 1 globally in time applications to quasilinear hyperbolic systems arising in physics and mechanics particularly to the system describing the motion of the relativistic string in the minkowski space time r 1 n are also given

a quantum analogue of the dihedral action on grassmannians in recent work launois and lenagan have shown how to construct a cocycle twisting of the quantum grassmannian and an isomorphism of the twisted and untwisted algebras that sends a given quantum minor to the minor whose index set is permuted according to the n cycle c 1 2 cdots n up to a power of q this twisting is needed because c does not naturally induce an automorphism of the quantum grassmannian as it does classically and semi classically we extend this construction to give a quantum analogue of the action on the grassmannian of the dihedral subgroup of s n generated by c and w 0 the longest element and this analogue takes the form of a groupoid we show that there is an induced action of this subgroup on the torus invariant prime ideals of the quantum grassmannian and also show that this subgroup acts on the totally nonnegative and totally positive grassmannians then we see that this dihedral subgroup action exists classically semi classically by poisson automorphisms and anti automorphisms a result of yakimov and in the quantum and nonnegative settings

primitive ideals of c q sl 3 the primitive ideals of the hopf algebrac q sl 3 are classified in particular it is shown that the orbits in primc q sl 3 under the action of the representation grouph c c are parameterized naturally byw w wherew is the associated weyl group it is shown that there is a natural one to one correspondence between primitive ideals ofc q sl 3 and symplectic leaves of the associated poisson algebraic groupsl 3 c

initial boundary value problem for the equation of timelike extremal surfaces in minkowski space this paper investigate the mixed initial boundary value problem for the equation of timelike extremal surfaces in minkowski space r1 1 n in the first quadrant under the assumptions that the initial data are bounded and the boundary data are small we prove the global existence and uniqueness of the c2 solutions of the initial boundary value problem for this kind of equation based on the existence results on global classical solutions we also show that as t tends to infinity the first order derivatives of the solutions approach c1 traveling wave under the appropriate conditions on the initial and boundary data geometrically this means the extremal surface approaches a generalized cylinder which is an exact solution

quantum graded algebras with a straightening law and the as cohen macaulay property for quantum determinantal rings and quantum grassmannians abstract we study quantum analogues of quotient varieties namely quantum grassmannians and quantum determinantal rings from the point of view of regularity conditions more precisely we show that these rings are as cohen macaulay and determine which of them are as gorenstein our method is inspired by the one developed by de concini eisenbud and procesi in the commutative case thus we introduce and study the notion of a quantum graded algebra with a straightening law on a partially ordered set showing in particular that among such algebras those whose underlying poset is wonderful are as cohen macaulay then we prove that both quantum grassmannians and quantum determinantal rings are quantum graded algebras with a straightening law on a wonderful poset hence showing that they are as cohen macaulay in this last step we are led to introduce and study to some extent natural quantum analogues of schubert varieties

asymptotic behavior of global classical solutions to goursat problem of quasilinear hyperbolic systems abstract this work is concerned with the asymptotic behavior of global c 1 solutions of the goursat problem for quasilinear hyperbolic systems we prove that when t tends to the infinity the solutions approach a combination of c 1 traveling wave solutions at algebraic rate 1 t provided that the boundary data decay at the rate 1 t 1 where is a positive constant finally we give an application to the equation for motion of an elastic string

limit cycle bifurcations near a generalized homoclinic loop in piecewise smooth systems with a hyperbolic saddle on a switch line in this paper we suppose a planar piecewise hamiltonian system has a generalized homoclinic loop with a hyperbolic saddle on a switch line and assume that there are two families of periodic orbits on both sides of the loop under perturbations we first give the expansion of the first melnikov function near the loop then by using the first coefficients in the expansion we study the number of limit cycles bifurcating from the homoclinic loop

long time behavior of classical solutions to the generalized goursat problem of quasilinear hyperbolic systems abstract we investigate the existence of a global classical solution to the generalized goursat problem under some degenerate assumptions of boundary conditions we prove that the solution approaches a combination of lipschitz continuous and a piecewise c 1 traveling wave solution

global weakly discontinuous solutions for hyperbolic conservation laws in the presence of a boundary abstract this work is a continuation of our previous work in the present paper we study the mixed initial boundary value problem for general n n quasilinear hyperbolic systems of conservation laws with non linear boundary conditions in the half space t x t 0 x 0 under the assumption that each characteristic with positive velocity is linearly degenerate we prove the existence and uniqueness of global weakly discontinuous solution u u t x with small amplitude and this solution possesses a global structure similar to that of the self similar solution u u x t of the corresponding riemann problem some applications to quasilinear hyperbolic systems of conservation laws arising in physics and other disciplines particularly to the system describing the motion of the relativistic string in minkowski space r 1 n are also given

a note on the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems abstract this paper is concerned with the asymptotic behavior of global classical solutions of diagonalizable quasilinear hyperbolic systems with linearly degenerate characteristic fields on the basis of the existence result for the global classical solution we prove that when t tends to the infinity the solution approaches a combination of c 1 traveling wave solutions provided that the c 1 norm and the bv norm of the initial data are bounded but possibly large in contrast to former results obtained by liu and zhou j liu y zhou asymptotic behaviour of global classical solutions of diagonalizable quasilinear hyperbolic systems math methods appl sci 30 2007 479 500 ours do not require their assumption that the system is rich in the sense of serre applications include that to the one dimensional born infeld system arising in string theory and high energy physics