existence and multiplicity of solutions for an elliptic system with nonlinear boundary conditions abstract in this paper we prove the existence and multiplicity of solutions for the elliptic system u u v v in with nonlinear coupling at the smooth boundary given by u h u v h v on where is a bounded domain of r n with smooth boundary is the outer normal derivative the proofs are done under suitable assumptions on the hamiltonian and based on the local linking theorem and a multiple critical points theorem of the critical point theory

segre product of artin schelter regular algebras of dimension 2 and embeddings in quantum p3 s the segre embedding of p1 p1as a smooth quadricqin p3corresponds to the surjection of the four dimensional polynomial ring onto the segre productsof two copies of the homogeneous coordinate ring of p1 we study segre products of noncommutative algebras if in particularaandbare two copies of a quantum p1thens i ai kbi is a twisted homogeneous coordinate ring of the quadricq the main result of this paper is the classification of all embeddings of the segre product of two quantum planes into so called quantum p3 s these are the proj of artin schelter regular algebrasrof global dimension four with the hilbert series of a commutative polynomial ring and which map ontos ifris not a twist of a polynomial ring then the point scheme ofreither is the union of the quadricqwith a line or is only the quadricq in the first case ris a central extension of a three dimensional artin schelter regular algebra and a twist of an algebra mapping onto the commutative homogeneous coordinate ring ofq in the second case such an algebraris the first known example of a four dimensional artin schelter regular algebra which is not determined by its point scheme

multiple solutions for asymptotically linear elliptic systems in this article we consider cooperative and noncooperative elliptic systems that are asymptotically linear at infinity we obtain infinitely many solutions with small energy if the potential is even if the noncooperative system is resonant both at zero and at infinity then the number of nontrivial solutions depends on the dimension of the eigenspaces between resonant values

new linking theorem and elliptic systems with nonlinear boundary condition in this paper we establish a new linking theorem without ps type assumption the new theorem produces bounded ps sequences the abstract result will be used to study the existence of nontrivial positive solutions to the elliptic system u u v v in with nonlinear boundary condition u gv u gu on where is a bounded domain of rn with smooth boundary is the outer normal derivative we consider weak superlinear and asymptotically linear cases

letter to the editor a note on the preconditioned gauss seidel gs method for linear systems wen li j comput appl math 182 2005 81 90 asserted that there are some errors in article by hiroshi niki kyouji harada munenori morimoto and michio sakakihara j comput appl math 164 165 2004 587 600 and li presented a new proof for the corresponding results in h niki et al in this paper we point out some errors in li s assertion moreover we show that a new proof presented by li is imperfect

noncommutative projective schemes an analogue of the concept of projective scheme is defined for noncommutative n graded algebras using the quotient category c of graded right a modules modulo its full subcategory of torsion modules we define proj a c a s where a is the object corresponding to the module aa and s is the autoequivalence defined by the shift of degrees the triples equivalent to proj a for a right noetherian graded algebra a are characterized in terms of a condition on extensions

on critical point theory for indefinite functionals in the presence of symmetries we consider functionals which are not bounded from above or from below even modulo compact perturbations and which exhibit certain symmetries with respect to the action of a compact lie group we develop a method which permits us to prove the existence of multiple critical points for such functionals the proofs are carried out directly in an infinite dimensional hilbert space and they are based on minimax arguments the applications given here are to hamiltonian systems of ordinary differential equations where the existence of multiple time periodic solutions is established for several classes of hamiltonians symmetry properties of these hamiltonians such as time translation invariancy or evenness are exploited introduction many variational problems which arise from physics or mathematical physics are indefinite in the sense that the functionals involved are not bounded from below or from above however some of these functionals defined in an appropriate function space are semidefinite in the sense that they are bounded from below or from above if perturbed with a weakly continuous functional this paper deals with functionals which are not semidefinite usually problems involving indefinite functionals are more difficult to handle and only very recently a method has been developed which permits us to treat such problems directly in an infinite dimensional function space 7 in 7 some theorems have been proved which establish the existence of at least one nontrivial critical point for such functionals in many physical situations there are problems which have symmetries with respect to the action of some lie group in this case we expect the existence of many critical points this has been established for semidefinite functionals cf e g 8 and 3 for even functionals 9 for a zp action with p prime numbers 15 16 10 and 6 for an s action in this paper we develop a method which allows us to estimate the number of critical points of indefinite functionals in the presence of symmetries more precisely received by the editors july 16 1980 and in revised form june 26 1981 1980 mathematics subject classification primary 58e05 70h05 secondary 34c25

a note on the preconditioner pm i sm kotakemori et al h kotakemori k harada m morimoto h niki a comparison theorem for the iterative method with the preconditioner i s m a x journal of computational and applied mathematics 145 2002 373 378 have reported that the convergence rate of the iterative method with a preconditioner p m i s m was superior to one of the modified gauss seidel method under the condition these authors derived a theorem comparing the gauss seidel method with the proposed method however through application of a counter example wen li wen li a note on the preconditioned gaussseidel gs method for linear systems journal of computational and applied mathematics 182 2005 81 91 pointed out that there exists a special matrix that does not satisfy this comparison theorem in this note we analyze the reason why such a to counter example may be produced and propose a preconditioner to overcome this problem

the noncommutative schemes of generalized weyl algebras abstract the first weyl algebra over k a 1 k x y x y y x 1 admits a natural z grading by letting deg x 1 and deg y 1 smith showed that gr a 1 is equivalent to the category of quasicoherent sheaves on a certain quotient stack using autoequivalences of gr a 1 smith constructed a commutative ring c graded by finite subsets of the integers he then showed gr a 1 gr c z fin in this paper we prove analogues of smith s results by using autoequivalences of a graded module category to construct rings with equivalent graded module categories for certain generalized weyl algebras we use autoequivalences defined in a companion paper so that these constructions yield commutative rings

on superquadratic elliptic systems in this article we study the existence of solutions for the elliptic system updelta u frac partial h partial v left u v x right text in upomega updelta v frac partial h partial v left u v x right text in upomega u 0 v 0 text on partial upomega where upomega is a bounded open subset of mathbb r n with smooth boundary partial upomega and the function h mathbb r 2 times overline upomega to mathbb r is of class c 1

rings graded equivalent to the weyl algebra abstract we consider the first weyl algebra a in the euler gradation and completely classify graded rings b that are graded equivalent to a that is the categories gr a and gr b are equivalent this includes some surprising examples in particular we show that a is graded equivalent to an idealizer in a localization of a we obtain this classification as an application of a general morita type characterization of equivalences of graded module categories given a z graded ring r an autoequivalence f of gr r and a finitely generated graded projective right r module p we show how to construct a twisted endomorphism ring end r f p and prove theorem the z graded rings r and s are graded equivalent if and only if there are an autoequivalence f of gr r and a finitely generated graded projective right r module p such that the modules f n p generate gr r and s end r f p

homoclinic orbits for asymptotically linear hamiltonian systems we study the existence of homoclinic orbits for first order time dependent hamiltonian systems z jhz z t where h z t depends periodically on t and hz z t is asymptotically linear in z as z we also consider an asymptotically linear schrodinger equation in rn

symmetry in the vanishing of ext over stably symmetric algebras abstract a frobenius algebra over a field k is called symmetric if the nakayama automorphism is an inner automorphism a stably symmetric algebra is defined to be a generalization of a symmetric k algebra in this paper we will study symmetry in the vanishing of ext for such algebras r namely for all finitely generated r modules m and n ext r i m n 0 for all i 0 if and only if ext r i n m 0 for all i 0 we show that a certain class of noetherian stably symmetric gorenstein algebras such as the group algebra of a finite group and the exterior algebra k n when n is odd have this symmetry using serre duality we also show that every exterior algebra k n whether n is even or odd has this symmetry for graded modules using koszul duality

a variational approach to subquadratic perturbations of elliptic systems abstract we consider subquadratic perturbations of semilinear elliptic systems which are in variational form a unified approach to both the cooperative and noncooperative problems is presented and a condition of nonquadraticity at infinity is introduced to treat the resonant cases through minimax techniques

a comparison theorem for the iterative method with the preconditioner i s max in 1991 gunawardena et al linear algebra appl 154 156 1991 123 have reported the modified gauss seidel method with a preconditioner i s in this article we propose to use a preconditioner i smax instead of i s here smax is constructed by only the largest element at each row of the upper triangular part of a by using the lemma established neumann and plemmons linear algebra appl 88 89 1987 559 we get the comparison theorem for the proposed method simple numerical examples are also given

improving the modified gauss seidel method for z matrices abstract in 1991 a d gunawardena et al reported that the convergence rate of the gauss seidel method with a preconditioning matrix i s is superior to that of the basic iterative method in this paper we use the preconditioning matrix i s if a coefficient matrix a is an irreducibly diagonally dominant z matrix then i s a is also a strictly diagonally dominant z matrix it is shown that the proposed method is also superior to other iterative methods

variational elliptic problems which are nonquadratic at infinity p co if n 1 2 then it is well known that the weak solutions u e h 2 of p are the critical points of the c functional j u ulvul dx jcf x u dx where f x s jif x t dt in this variational setting the literature usually distinguishes between two situations for prob lem p the subquadratic situation where the potential f satisfies lim suplsl f x s s2 i c m and the superquadratic situation where f satisfies limlsl m f x s s2 00 the main goal of this paper is to present a unified approach to both situations by means of a condition of nonquadraticity at infinity on f more precisely our hypotheses are as follows

modified iterative methods for consistent linear systems abstract in order to solve a linear system ax b certain elementary row operations are performed on a before applying the gauss seidel or jacobi iterative methods it is shown that when a is a nonsingular m matrix or a singular tridiagonal m matrix the modified method yields considerable improvement in the rate of convergence for the iterative method it is also shown that in some cases this method is superior to certain other modified iterative methods the performance of this modified method on some matrices other than m matrices is also investigated

some comparison theorems for weak nonnegative splittings of bounded operators abstract the comparison of the asymptotic rates of convergence of two iteration matrices induced by two splittings of the same matrix has arisen in the works of many authors in this paper we derive new comparison theorems for weak nonnegative splittings and weak splittings of bounded operators in a general banach space and rather general cones and in a hilbert space which extend some of the results obtained by wo nicki japan j indust appl math 11 1994 289 342 and marek and szyld numer math 44 1984 23 35 furthermore we present new theorems also for bounded operator which extend some results by csordas and varga numer math 44 1984 23 35 for weak nonnegative splittings of matrices

the picard group of the graded module category of a generalized weyl algebra abstract the first weyl algebra a 1 k x y x y y x 1 is naturally z graded by letting deg x 1 and deg y 1 sue sierra studied gr a 1 category of graded right a 1 modules computing its picard group and classifying all rings graded equivalent to a 1 in this paper we generalize these results by studying the graded module category of certain generalized weyl algebras we show that for a generalized weyl algebra a f with base ring k z defined by a quadratic polynomial f the picard group of gr a f is isomorphic to the picard group of gr a 1

a quotient stack related to the weyl algebra abstract let a denote the ring of differential operators on the affine line with its two usual generators t and d d t given degrees 1 and 1 respectively let x be the stack having coarse moduli space the affine line spec k z and isotropy groups z 2 at each integer point then the category of graded a modules is equivalent to the category of quasi coherent sheaves on x

the dynamics of leavitt path algebras abstract recently it was shown that the notion of flow equivalence of shifts of finite type in symbolic dynamics is related to the morita theory and the grothendieck group in the theory of leavitt path algebras abrams et al 2011 4 in this paper we show that the notion of the conjugacy of shifts of finite type is closely related to the graded morita theory and consequently the graded grothendieck group this fits into the general framework we have in these two theories conjugacy yields the flow equivalence and the graded morita equivalence can be lifted to the morita equivalence starting from a finite directed graph the observation that the graded grothendieck group of the leavitt path algebra associated to e coincides with the krieger dimension group of the shift of finite type associated to e provides a link between the theory of leavitt path algebras and symbolic dynamics it has been conjectured that the ordered graded grothendieck group as z x x 1 module we call this the graded dimension group classifies the unital leavitt path algebras completely hazrat 2013 20 via the above correspondence utilising the results from symbolic dynamics we prove that for two purely infinite simple unital leavitt path algebras if their graded dimension groups are isomorphic then the algebras are isomorphic

stationary splitting iterative methods for the matrix equation axb c abstract stationary splitting iterative methods for solving a x b c are considered in this paper the main tool to derive our new method is the induced splitting of a given nonsingular matrix a m n by a matrix h such that i h 1 exists convergence properties of the proposed method are discussed and numerical experiments are presented to illustrate its computational efficiency and the effectiveness of some preconditioned variants in particular for certain surface fitting applications our method is much more efficient than the progressive iterative approximation pia a conventional iterative method often used in computer aided geometric design cagd

comparison theorems for weak splittings of bounded operators comparison theorems for weak splittings of bounded operators are presented these theorems extend the classical comparison theorem for regular splittings of matrices by varga the less known result by wo nicki and the recent results for regular and weak regular splittings of matrices by neumann and plemmons elsner and lanzkron rose and szyld the hypotheses of the theorems presented here are weaker and the theorems hold for general banach spaces and rather general cones hypotheses are given which provide strict inequalities for the comparisons it is also shown that the comparison theorem by alefeld and volkmann applies exclusively to monotone sequences of iterates and is not equivalent to the comparison of the spectral radius of the iteration operators

a note on the preconditioned gauss seidal gs method for linear systems in this note recent comparison results for preconditioned gauss seidel gs methods are discussed a new strict comparison result between two different preconditioned gs methods is given some errors in a recent article by niki et al j comput appl math 164 165 2004 587 are pointed out and a new proof for the corresponding results in niki et al is presented

differential systems with strongly indefinite variational structure abstract the euler lagrange equations of lagrangians with a strongly indefinite quadratic part are studied by means of the direct min max method of benci and rabinowitz the simplest case is a system of two coupled semilinear poisson equations using an analytic framework of a suitable family of products of fractional sobolev spaces existence results are obtained for all nonlinearities with subcritical growth rates

the survey of preconditioners used for accelerating the rate of convergence in the gauss seidel method several preconditioned iterative methods reported in the literature have been used for improving the convergence rate of the gauss seidel method in this article on the basis of nonnegative matrix comparisons between some splittings for such preconditioned matrices are derived simple numerical examples are also given