exponentially evolving densities the notion of a conserved density is generalized to those densities whose integrals evolve exponentially in the formal differential algebra used to find them only a slight modification of the conserved density case is needed various examples for quasilinear evolution equations are given and an application to finding higher order conserved densities by taking the poisson bracket of exponentially evolving densities is indicated

third order upwind finite element solutions of high reynolds number flows abstract a full upwind finite element scheme based on the petrov galerkin formulation is presented for accurate solutions of high reynolds number flows and then a simplified finite scheme with third order upwinding is developed in order to obtain a practical algorithm for numerical simulation because structures of the full upwind finite element scheme are very complicated numerical dissipation added in the proposed upwind scheme is represented by high order derivatives of flow velocities and pressure computational results for the subcritical and the supercritical reynolds number flows around a circular cylinder and for flows around twin circular cylinders are shown to illustrate the effectiveness and robustness of the simplified upwind finite element scheme

category equivalences involving graded modules over path algebras of quivers abstract let q be a finite quiver with vertex set i and arrow set q 1 k a field and k q its path algebra with its standard grading this paper proves some category equivalences involving the quotient category qgr k q gr k q fdim k q of graded k q modules modulo those that are the sum of their finite dimensional submodules namely qgr k q mod s q gr l q mod l q 0 qgr k q n here s q lim end k i k q 1 n is a direct limit of finite dimensional semisimple algebras q is the quiver without sources or sinks that is obtained by repeatedly removing all sinks and sources from q l q is the leavitt path algebra of q l q 0 is its degree zero component and q n is the quiver whose incidence matrix is the n th power of that for q it is also shown that all short exact sequences in qgr k q the full subcategory of finitely presented objects in qgr k q split consequently qgr k q can be given the structure of a triangulated category with suspension functor the serre degree twist 1 this triangulated category is equivalent to the singularity category d b d perf where is the radical square zero algebra k q k q 2 and d b is the bounded derived category of finite dimensional left modules

an infinity of conservation laws for highly symmetric nonisentropic ideal gas flow abstract it is shown that for nonisentropic gas flow which is either a one dimensional plane flow or a flow with cylindrical or spherical symmetry there exists an infinity of local conservation laws from these for appropriate initial conditions an infinity of global conserved quantities can be derived

infinitely many commuting symmetries and constants of motion in involution for explicitly time dependent evolution equations the existing symmetry approach to exactly solvable time evolution equations in one spatial dimension is extended to the case that the equation depends explicitly on the time and space coordinates in this context commuting symmetries constants of motion in involution strong and hereditary symmetries squared eigenfunction operators and hamiltonian structures are discussed the cylindrical korteweg de vries equation is used as an illustrative example

the leavitt path algebra of a graph abstract for any row finite graph e and any field k we construct the leavitt path algebra l e having coefficients in k when k is the field of complex numbers then l e is the algebraic analog of the cuntz krieger algebra c e described in i raeburn graph algebras in cbms reg conf ser math vol 103 amer math soc 2005 the matrix rings m n k and the leavitt algebras l 1 n appear as algebras of the form l e for various graphs e in our main result we give necessary and sufficient conditions on e which imply that l e is simple

multi symplectic lagrangian one dimensional gas dynamics the equations of lagrangian ideal one dimensional compressible gas dynamics are written in a multi symplectic form using the lagrangian mass coordinate m and time t as independent variables and in which the eulerian position of the fluid element x x m t is one of the dependent variables this approach differs from the eulerian multi symplectic approach using clebsch variables lagrangian constraints are used to specify equations for xm xt and st consistent with the lagrangian map where s is the entropy of the gas we require st 0 corresponding to advection of the entropy s with the flow we show that the lagrangian hamiltonian equations are related to the de donder weyl multi momentum formulation the pullback conservation laws and the symplecticity conservation laws are discussed the pullback conservation laws correspond to invariance of the action with respect to translations in time energy conservation and translations in m in noether s theorem the conservation law due to m translatio

factorization of operators ii we extend the methods of a previous paper j math phys 21 2508 1980 factorizing the general scalar third order differential operator and obtain a miura transformation for the boussinesq equation we give a general factorized eigenvalue problem we also give a hamiltonian structure associated with the factorized eigenvalue problem we derive several isospectral flows some of klein gordon type

an infinite lie group of symmetry of one dimensional gas flow for a class of entropy distributions abstract we have shown in earlier works the existence of three previously unknown symmetries of the equations of one dimensional gas dynamics with arbitrary entropy distribution and arbitrary polytropic index these symmetries are seen here to form a group whenever the equation of state is of the form p 3 a0 a1m a2m2 2 where m dr is the lagrangian mass coordinate introducing the remaining symmetry of space translation enlarges the group into a lie group of symmetry of infinite order from which an infinite number of conservation laws can be deduced by application of noether s theorem the lie group has a finite sub algebra of order eight which has su3 structure the list of associated conservation laws includes each of the six ones that are derivable from general physical principles namely the energy momentum and the center of mass integrals two integrals expressing scale invariance and one associated with the virial theorem the remaining two integrals of the octet are of a new type such a situation reminds us of the case of the korteweg de vries equation in the soliton problem where the symmetries and infinite number of conservation laws arise as a result of the possibility to linearize through the inverse scattering method thus the question is raised of whether the inverse scattering method also applies to gas dynamical equations with the above equation of state or else whether another method of linearization may be found

the hamiltonian structure of the maxwell vlasov equations morrison 25 has observed that the maxwell vlasov and poisson vlasov equations for a collisionless plasma can be written in hamiltonian form relative to a certain poisson bracket we derive another poisson structure for these equations by using general methods of symplectic geometry the main ingredients in our construction are the symplectic structure on the co adjoint orbits for the group of canonical transformations and the symplectic structure for the phase space of the electromagnetic field regarded as a gauge theory our poisson bracket satisfies the jacobi identity whereas morrison s does not 37 our construction also shows where canonical variables can be found and can be applied to the yang mills vlasov equations and to electromagnetic fluid dynamics

the hamiltonian structures of the nonlinear schrodinger equation in the classical limit using madelung s hydrodynamical variables it is shown that the bi hamiltonian structure of the nonlinear schrodinger equation goes over to a bi hamiltonian structure of the shallow water wave equations in the classical limit

conservation laws generated by pairs of non hamiltonian symmetries we consider systems of partial differential equations of the form w here u u u are dependent variables and k is a q tuple of partial differential functions i e smooth functions of x or x and the various derivatives z 8ui i jr j 8 3 ap ai 3 l the system is autonomous k does not depend on the time t explicitly our point of view is that of the formal variational calculus as in gel fand and dikii i 4 gel fand and dorfman 5 and olver 7 io the essential elements of this calculus will be set out in section 2 a scalar partial differential function t which may also depend explicitly on time is conserved under the flow of u k if i t dx is independent of time ipp this is formally equivalent to d t a divergence expression say d p d d where the pi are partial functions and di and d are total derivatives for example whereas d tu 2tuk uj

a new finite element formulation for computational fluid dynamics viii the galerkin least squares method for advective diffusive equations abstract galerkin least squares finite element methods are presented for advective diffusive equations galerkin least squares represents a conceptual simplification of supg and is in fact applicable to a wide variety of other problem types a convergence analysis and error estimates are presented

noether type conservation laws for perfect fluid motions a general approach is developed for the derivation of conservation laws in continuum physics a noether type theorem is applied in connection with transformations which leave the action functional invariant to within the integral of a divergence specific results are derived in the case of fluid dynamics the pertinent equations are considered within the lagrangian material description and are associated with a genuine variation formulation the physical meaning of the conservation laws is emphasized and the greater generality of the approach is commented upon

third order upwind finite element computation of the incompressible navier stokes equations part ii aerodynamic characteristics of a rectangular cylinder with an angle of attack abstract aerodynamic characteristics of rectangular cylinders at depth breadth ratios of 1 and 2 are investigated in detail by numerical simulation of the incompressible navier stokes equations using a third order upwind finite element scheme in the numerical computation a uniform flow is given at the inlet boundary in a computational region and the rectangular cylinders are inclined toward the approaching flow in the range of angles of attack from 0 up to 45 the present solutions for aerodynamic forces and moment by forces of the inclined rectangular cylinders are compared with other numerical results and experimental results

conservation laws and symmetries of generalized sine gordon equations we study some systems of non linear pde s eqs 1 1 below which can be regarded either as generalizations of the sine gordon equation or as two dimensional versions of the toda lattice equations we show that these systems have an infinite number of non trivial conservation laws and an infinite number of symmetries the second result is deduced from the first by a variant of the hamiltonian formalism for evolution equations we also consider some specializations of the systems

multi lagrangians for integrable systems we propose a general scheme to construct multiple lagrangians for completely integrable nonlinear evolution equations that admit multi hamiltonian structure the recursion operator plays a fundamental role in this construction we use a conserved quantity higher lower than the hamiltonian in the potential part of the new lagrangian and determine the corresponding kinetic terms by generating the appropriate momentum map this leads to some remarkable new developments we show that nonlinear evolutionary systems that admit n fold first order local hamiltonian structure can be cast into variational form with 2n 1 lagrangians which will be local functionals of clebsch potentials this number increases to 3n 2 when the miura transformation is invertible furthermore we construct a new lagrangian for polytropic gas dynamics in 1 1 dimensions which is a free local functional of the physical field variables namely density and velocity thus dispensing with the necessity of introducing clebsch potentials entirely this is a consequence of bi hamiltonian structure with a compatible pair of first and third order hamiltonian operators derived from sheftel s recursion operator

boundary conditions and the first order conserved density for gas dynamics boundary conditions for gas dynamics are chosen to make the first order conserved density finitely integrable the action of the corresponding third order noetherian symmetry on the boundary condition is discussed a generalisation to first order diagonalised quasilinear systems is considered

exact symmetries of unidimensional self similar flow in addition to the symmetries that are known to apply to arbitrary flow the self similar equations may present other symmetries of their own we present here such a symmetry of the self similar unidimensional flow of an adiabatic inviscid fluid with arbitrary polytropic index and arbitrary power law entropy distribution the new symmetry can be extended to the non self similar case if the flow is assumed isentropic a connection with the theory of riemann invariants is also discussed

a new finite element formulation for computational fluid dynamics x the compressible euler and navier stokes equations a space time element method is presented for solving the compressible euler and navier stokes equations the proposed formulation includes the variational equation predictor multi corrector algorithms and boundary conditions the variational equation is based on the time discontinuous galerkin method in which the physical entropy variables are employed a least squares operator and a discontinuity capturing operator are added resulting in a high order accurate and unconditionally stable method implicit explicit predictor multi corrector algorithms applicable to steady as well as unsteady problems are presented techniques are developed to enhance their efficiency implementation of boundary conditions is addressed in particular a technique is introduced to satisfy nonlinear essential boundary conditions and a consistent method is presented to calculate boundary fluxes numerical results are presented to demonstrate the performance of the method

su 3 symmetry of the equations of unidimensional gas flow with arbitrary entropy distribution we have shown in an earlier work that assuming a particular class of equations of state the euler equations of one dimensional gas flow are invariant under an su 3 group of transformations and in fact admit of a lie group of symmetry of infinite order they therefore possess an infinite number of conservation laws we show in the present work that the su 3 symmetrical formalism still brings about tremendous simplification and analytical order in the most general case where the equation of state is arbitrary the six characteristic equations assume a vector form and relate two conjugate three dimensional vectors u and x the su 3 symmetry is only broken to a minor extent through the occurrence of a multiplicative factor in the equations the conservation laws take the form of the cauchy integrability condition for the elements of a traceless second rank tensor eij and taken all together form an su 3 octet in the most general case however there exist four conservation laws only five if the

a nonlinear hamiltonian structure for the euler equations abstract the euler equations for inviscid incompressible fluid flow have a hamiltonian structure in eulerian coordinates the hamiltonian operator though depending on the vorticity conservation laws arise from two sources one parameter symmetry groups which are completely classified yield the invariance of energy and linear and angular momenta degeneracies of the hamiltonian operator lead in three dimensions to the total helicity invariant and in two dimensions to the area integrals reflecting the point wise conservation of vorticity it is conjectured that no further conservation laws exist indicating that the euler equations are not completely integrable in particular do not have soliton like solutions

on graded irreducible representations of leavitt path algebras using the e algebraic branching systems various graded irreducible representations of a leavitt path k algebra l of a directed graph e are constructed the concept of a laurent vertex is introduced and it is shown that the minimal graded left ideals of l are generated by the laurent vertices or the line points leading to a detailed description of the graded socle of l following this a complete characterization is obtained of the leavitt path algebras over which every graded irreducible representation is finitely presented a useful result is that the irreducible representation v p induced by infinite paths tail equivalent to an infinite path p we call this a chen simple module is graded if and only if p is an irrational path we also show that every one sided ideal of l is graded if and only if the graph e contains no cycles supplementing the theorem of one of the co authors that every leavitt path algebra l is graded von neumann regular we show that l is graded self injective if and only if l is a graded semi simple algebra made up of matrix rings of arbitrary size over the field k or the graded field k xn x n where n n

finitely presented simple modules over leavitt path algebras abstract let e be an arbitrary graph and k be any field we construct various classes of non isomorphic simple modules over the leavitt path algebra l k e induced by vertices which are infinite emitters closed paths which are exclusive cycles and paths which are infinite and call these simple modules chen modules it is shown that every primitive ideal of l k e can be realized as the annihilator ideal of some chen module our main result establishes the equivalence between a graph theoretic condition and various conditions concerning the structure of simple modules over l k e

leavitt path algebras with finitely presented irreducible representations abstract let e be an arbitrary graph k be any field and let l l k e be the corresponding leavitt path algebra necessary and sufficient conditions both graphical and algebraic are given under which all the irreducible representations of l are finitely presented in this case the graph e turns out to be row finite and the cycles in e form an artinian partial ordered set under a defined relation when the graph is e is finite the above graphical conditions were shown in 6 to be equivalent to l k e having finite gelfand kirillov dimension examples are constructed showing that this equivalence no longer holds for infinite graphs and a complete description is obtained of leavitt path algebras over arbitrary graphs having finite gelfand kirillov dimensions the building blocks for these algebras seem to be von neumann rings and the laurent polynomial ring k x x 1

finite element solution of the unsteady navier stokes equations by a fractional step method abstract a finite element method is presented for the numerical simulation of time dependent incompressible viscous flows the method is based on a fractional step approach to the time integration of the navier stokes equations in which only the incompressibility condition is treated implicitly this leads to a computational scheme of extremely simple algorithmic structure that is particularly attractive for cost effective solutions of large scale problems numerical results indicate the versatility and effectiveness of the proposed method

common structure of several completely integrable non linear equations the author consider several non linear equations which are known to be completely integrable by the method of inverse scattering transform ranging from sine gordon and the dodd bullough equation 1976 to the axisymmetric stationary einstein maxwell equations such equations are known to be invariant under a fundamental lie group of symmetry g such as sl 2 or sl 3 in the simpler cases the author s treatment starts with the reformulation of the given equations as a system involving the scalar invariants of the lie group g only this scalar formulation is not only g invariant but also manifestly invariant under the continuous group associated with the emergence of a spectral parameter and is thus expected to be more fundamental and simpler in all the cases that the author has considered including the einstein maxwell system vector pseudopotentials of dimension not higher than three have been derived backlund transformations have been found which generate multisolitons starting from the vacuum and have the general reciprocal form x 1 x where x is a component of one of the vector or tensor pseudopotentials the author also mentions the result that kinnersley s formulation 1977 of the ernst vacuum equation 1968 is formally equivalent to the fluid dynamical problem of unidimensional ideal gas flow the dodd bullough equation is equivalent to another particular case of gas motion

higher order symmetries of the compressible one dimensional isentropic fluid equations the equations of compressible one dimensional isentropic flow are shown to have infinitely many higher order symmetries by means of recursion operators furthermore a new first order conserved functional is found to be associated with a third order symmetry via noether s theorem

on simple modules over leavitt path algebras abstract given an arbitrary graph e and any field k a new class of simple modules over the leavitt path algebra l k e is constructed by using vertices that emit infinitely many edges in e the corresponding annihilating primitive ideals are also described given a fixed simple l k e module s we compute the cardinality of the set of all simple l k e modules isomorphic to s using a boolean subring of idempotents induced by paths in e bounds for the cardinality of the set of distinct isomorphism classes of simple l k e modules are given we also obtain a complete structure theorem about the leavitt path algebra l k e of a finite graph e over which every simple module is finitely presented

multi hamiltonian structure of equations of hydrodynamic type the discussion of the hamiltonian structure of two component equations of hydrodynamic type is completed by presenting the hamiltonian operators for euler s equation governing the motion of plane sound waves of finite amplitude and another quasilinear second order wave equation there exists a doubly infinite family of conserved hamiltonians for the equations of gas dynamics that degenerate into one namely the benney sequence for shallow water waves infinite sequences of conserved quantities for these equations are also presented in the case of multicomponent equations of hydrodynamic type it is shown that kodama s generalization of the shallow water equations admits bi hamiltonian structure

structure of leavitt path algebras of polynomial growth we determine the structure of leavitt path algebras of polynomial growth and discuss their automorphisms and involutions

uniqueness theorems and ideal structure for leavitt path algebras abstract we prove leavitt path algebra versions of the two uniqueness theorems of graph c algebras we use these uniqueness theorems to analyze the ideal structure of leavitt path algebras and give necessary and sufficient conditions for their simplicity we also use these results to give a proof of the fact that for any graph e the leavitt path algebra l c e embeds as a dense subalgebra of the graph c algebra c e this embedding has consequences for graph c algebras and we discuss how we obtain new information concerning the construction of c e

on the use of isospectral eigenvalue problems for obtaining hereditary symmetries for hamiltonian systems we present an algorithmic method for obtaining an hereditary symmetry the generalized squared eigenfunction operator from a given isospectral eigenvalue problem this method is applied to the n n eigenvalue problem considered by ablowitz and haberman and to the eigenvalue problem considered by alonso the relevant hamiltonian formulations are also determined finally an alternative method is presented in the case two evolution equations are related by a miura type transformation and their hamiltonian formulations are known

third order upwind finite element computation of the incompressible navier stokes equations part i computation of flow around rectangular cylinders abstract aerodynamic behaviours of two dimensional unsteady flows around rectangular cylinders are numerically investigated by third order upwind finite element computation of the incompressible navier stokes equations a turbulence model such as the two equation model is not incorporated in the computational scheme so that the rectangular cylinders at depth breadth ratios from 0 7 to 4 to be considered are located in a uniform flow it is found that the numerical results of flows around the rectangular cylinders obtained by using the third order upwind finite element scheme are in good agreement with other computational and experimental results

conservation laws for shallow water waves on a sloping beach shallow water waves are governed by a pair of nonlinear partial differential equations we transfer the associated homogeneous and nonhomogeneous systems corresponding to constant and sloping depth respectively to the hodograph plane where we find all the nonsimple wave solutions and construct infinitely many polynomial conversation laws we also establish correspondence between conservation laws and hodograph solutions as well as backlund transformations by using the linear nature of the problems on the hodograph plane

the theory of prime ideals of leavitt path algebras over arbitrary graphs given an arbitrary graph e and a field k the prime ideals as well as the primitive ideals of the leavitt path algebra lk e are completely characterized in terms of their generators the stratification of the prime spectrum of lk e is indicated with information on its individual stratum necessary and sufficient conditions are given on the graph e under which every prime ideal of the leavitt path algebra lk e is primitive leavitt path algebras with krull dimension zero are characterized and those with various prescribed krull dimension are constructed the minimal prime ideals of lk e are described in terms of the graphical properties of e and using this complete descriptions of the height one as well as the co height one prime ideals of lk e are given

symmetries and a hierarchy of the general kdv equation two groups old and new of symmetries and their lie algebra properties for the kdv and cylindrical kdv equations are unified and extended to the general kdv equation

symmetries and a hierarchy of the general modified kdv equation two groups of symmetries and their lie algebra properties for the modified kdv equation are extended to the general modified kdv equation and the miura transformation between the general kdv equation and the general modified equation is also established

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100