one parameter lie groups and inverse integrating factors of n th order autonomous systems abstract the method of obtaining inverse integrating factors of n th order autonomous systems using one parameter lie groups admitted by the systems is given by describing n th order autonomous systems with differential form systems the properties of inverse integrating factors of the n th order autonomous systems are presented

on the geometry of positive maps in matrix algebras ii abstract geometric relations among the identity map the transpose map and the map x 1 n x tr x 1 n in the matrix algebra mn are illustrated providing parametric examples of maps which are not k 1 positive but which satisfy the matrix schwartz inequality of order k for 1 k n 1

a new criterion for indecomposability of positive maps we give a criterion for a positive mapping on the space of operators on a hilbert space to be indecomposable we show that this criterion can be applied to two families of positive maps these families of maps can then be used to form separability criteria for bipartite quantum states that can detect the entanglement of bound entangled quantum states

convergence result for the constraint preserving mid point scheme for micromagnetism an important progress was recently done in numerical approximation of weak solutions to a micromagnetic model equation the problem with the nonconvex side constraint of preserving the length of the magnetization was tackled by using reduced integration several schemes were proposed and their convergence to weak solutions was proved all schemes were derived from the landau lifshitz gilbert form of the micromagnetic equation however when the precessional term in the original landau lifshitz ll form of the micromagnetic equation tends to zero the above schemes become unusable we propose a scheme derived from the mid point rule for the ll form of the micromagnetic equation combined with the reduced integration we show convergence to a weak solution of the ll equation and demonstrate the usefulness of the proposed scheme to study the limit process when the precessional parameter of the micromagnetic equation goes to zero

generalized choi maps in three dimensional matrix algebra we consider a class of positive linear maps in the three dimensional matrix algebra which are generalizations of the positive linear map constructed by choi in the relation with positive semidefinite biquadratic forms we find conditions for which such maps are completely positive completely copositive decomposable and two positive

cones of positive maps and their duality relations the structure of cones of positive and k positive maps acting on a finite dimensional hilbert space is investigated special emphasis is given to their duality relations to the sets of superpositive and k superpositive maps we characterize k positive and k superpositive maps with regard to their properties under taking compositions a number of results obtained for maps are also rephrased for the corresponding cones of block positive k block positive separable and k entangled operators due to the jamiolkowski choi isomorphism generalizations to a situation where no such simple isomorphism is available are also made employing the idea of mapping cones as a side result to our discussion we show that extreme entanglement witnesses which are optimal should be of special interest in entanglement studies

cones with a mapping cone symmetry in the finite dimensional case abstract in the finite dimensional case we present a new approach to the theory of cones with a mapping cone symmetry first introduced by stormer our method is based on a definition of an inner product in the space of linear maps between two algebras of operators and the fact that the jamiolkowski choi isomorphism is an isometry we consider a slightly modified class of cones although not substantially different from the original mapping cones by stormer using the new approach several known results are proved faster and often in more generality than before for example the dual of a mapping cone turns out to be a mapping cone as well without any additional assumptions the main result of the paper is a characterization of cones with a mapping cone symmetry saying that a given map is an element of such cone if and only if the composition of the map with the conjugate of an arbitrary element in the dual cone is completely positive a similar result was known in the case where the map goes from an algebra of operators into itself and the cone is a symmetric mapping cone our result is proved without the additional assumptions of symmetry and equality between the domain and the target space we show how it gives a number of older results as a corollary including an exemplary application

extension of positive maps into b h soient 7b h et 7b k des espaces de hilbert complexes et soit a un sous espace lineaire auto adjoint norme clos d operateurs bornes b 7b k sur 7b k contenant l operateur identite si on note par b a 7b h les applications positives dans l ensemble b a 7b h des applications lineaires bornees de a dans b 7b h on etudie le probleme d extension pour des applications dans b a 7b h aux applications dans b b 7b k 7b h

three by three bound entanglement with general unextendible product bases we discuss the subject of unextendible product bases with the orthogonality condition dropped and we prove that the lowest rank non separable positive partial transpose states i e states of rank 4 in 3 3 systems are always locally equivalent to a projection onto the orthogonal complement of a linear subspace spanned by an orthogonal unextendible product basis the product vectors in the kernels of the states belong to a non zero measure subset of all general unextendible product bases nevertheless they can always be locally transformed to the orthogonal form this fully confirms the surprising numerical results recently reported by leinaas et al parts of the paper rely heavily on the use of bezout s theorem from algebraic geometry

on the nonexistence of rational first integrals for nonlinear systems and semiquasihomogeneous systems in this paper we first give a necessary condition for general nonlinear systems to have rational first integrals then by using the so called kowalevsky exponents we present a criterion for nonexistence of rational first integrals for semiquasihomogeneous systems

lie point symmetries partial noether operators and first integrals of the painleve gambier equations abstract we utilize the lie tresse linearization method to obtain linearizing point transformations of certain autonomous nonlinear second order ordinary differential equations contained in the painleve gambier classification these point transformations are constructed using the lie point symmetry generators admitted by the underlying painleve gambier equations it is also shown that those painleve gambier equations which have a few lie point symmetries and hence are not linearizable by this method can be integrated by a quadrature moreover by making use of the partial lagrangian approach we obtain time dependent and time independent first integrals for these painleve gambier equations which have not been reported in the earlier literature a comparison of the results obtained in this paper is made with the ones obtained using the generalized sundman linearization method

the mimetic finite difference method for the landau lifshitz equation the landau lifshitz equation describes the dynamics of the magnetization inside ferromagnetic materials this equation is highly nonlinear and has a non convex constraint the magnitude of the magnetization is constant which poses interesting challenges in developing numerical methods we develop and analyze explicit and implicit mimetic finite difference schemes for this equation these schemes work on general polytopal meshes which provide enormous flexibility to model magnetic devices with various shapes a projection on the unit sphere is used to preserve the magnitude of the magnetization we also provide a proof that shows the exchange energy is decreasing in certain conditions the developed schemes are tested on general meshes that include distorted and randomized meshes the numerical experiments include a test proposed by the national institute of standard and technology and a test showing formation of domain wall structures in a thin film

formal first integrals for periodic systems in the present paper we give an elementary proof for the result of li et al 2003 6 about nonexistence of formal first integrals for periodic systems in a neighborhood of a constant solution moreover we present a criterion about partial existence of formal first integrals for the periodic system by using the floquet s theory

choi matrices norms and entanglement associated with positive maps on matrix algebras abstract we study positive maps of b k into b h for finite dimensional hilbert spaces k and h our main emphasis is on how choi matrices and estimates of their norms with respect to mapping cones reflect various properties of the maps special attention will be given to entanglement properties and k positive maps

generation of mapping cones from small sets we answer in the affirmative a recently posed question that asked if there exists an untypical convex mapping cone i e one that does not arise from the transpose map and the cones of k positive and k superpositive maps we explicitly construct such a cone based on atomic positive maps our general technique is to consider the smallest convex mapping cone generated by a single map and we derive several results on such mapping cones we use this technique to also present several other examples of untypical mapping cones including a family of cones generated by spin factors we also provide a full characterization of mapping cones generated by single elements in the qubit case in terms of their typicality

an integrating factor matrix method to find first integrals in this paper we develop an integrating factor matrix method to derive conditions for the existence of first integrals we use this novel method to obtain first integrals along with the conditions for their existence for two and three dimensional lotka volterra systems with constant terms the results are compared to previous results obtained by other methods

on the first integrals of n th order autonomous systems the first integrals of n th order autonomous systems are considered by getting integrating factors using n 1 n 1 first integrals of the systems a necessary condition on the existence of global first integrals of n th order autonomous systems is presented it is also proved that n 1 n 1 functionally independent first integrals of an n th order autonomous system can be obtained if the system possesses n 1 n 1 functionally independent integrating factors based on one parameter lie groups admitted by n th order autonomous systems several methods to obtain first integrals of the systems are presented simultaneously several related examples are given to illustrate the feasibility and the effectiveness of the proposed method

cones and norms in the tensor product of matrix spaces abstract this paper is of expository nature in the tensor product of two matrix spaces m m and m n we consider three natural cones the first is the cone p 0 of positive semi definite matrices the second is the cone p generated by tensor products of positive semi definite matrices in the factor spaces and the third cone p is defined as the dual cone of p correspondingly in the tensor space identified with m mn besides the spectral and the trace norms we consider several tensor norms generated by sepectral and or trace norms in the factor spaces we will study relations among those norms as well as relations among the three kinds of order intervals in the tensor product

on the complete and partial integrability of non hamiltonian systems abstract the methods of singularity analysis are applied to several third order non hamiltonian systems of physical significance including the lotka volterra equations the three wave interaction and the rikitake dynamo model complete integrability is defined and new completely integrable systems are discovered by means of the painleve property in all these cases we obtain integrals which reduce the equations either to a final quadrature or to an irreducible second order ordinary differential equation ode solved by painleve transcendents relaxing the painleve property we find many partially integrable cases whose movable singularities are poles at leading order with in t t 0 terms entering at higher orders in an n th order generalized rossler model a precise relation is established between the partial fulfillment of the painleve conditions and the existence of n 2 integrals of the motion

the inverse integrating factor for some classes of n th order autonomous differential systems abstract the inverse integrating factor for some classes of higher order autonomous differential systems is studied in this paper some sufficient conditions on the existence of the inverse integrating factors of certain classes of n th order autonomous differential systems are presented also the explicit forms of the inverse integrating factors can be shown simultaneously some related examples are given to illustrate the feasibility and the effectiveness of the proposed results

the first integral method to study the burgers korteweg de vries equation in this paper applying the theory of commutative algebra we propose a new approach which we currently call the first integral method to study the burgers korteweg de vries equation

generalized rational first integrals of analytic differential systems abstract in this paper we mainly study the necessary conditions for the existence of functionally independent generalized rational first integrals of ordinary differential systems via the resonances the main results extend some of the previous related ones for instance the classical poincare s one poincare 1891 1897 16 the furta s one furta 1996 8 part of chen et al s ones chen et al 2008 4 and the shi s one shi 2007 18 the key point in the proof of our main results is that functionally independence of generalized rational functions implies the functionally independence of their lowest order rational homogeneous terms

there is no direct generalization of positive partial transpose criterion to the three by three case we show that there cannot exist a straightforward generalization of the famous positive partial transpose criterion to three by three systems we call straightforward generalizations that use a finite set of positive maps and arbitrary local rotations of the tested two partite state in particular we show that a family of extreme positive maps discussed in a paper by ha and kye open syst inf dyn 18 323 337 2011 cannot be replaced by a finite set of witnesses in the task of entanglement detection in three by three systems in a more mathematically elegant parlance our result says that the convex cone of positive maps of the set of three dimensional matrices into itself is not finitely generated as a mapping cone

indecomposable positive maps in low dimensional matrix algebras abstract we classify a series of positive maps in low dimensional matrix algebras with respect to the degree of indecomposability and provide stronger indecomposable maps in m n we also present a basic result about the atomic property

unextendible product bases and bound entanglement an unextendible product basis upb for a multipartite quantum system is an incomplete orthogonal product basis whose complementary subspace contains no product state we give examples of upbs and show that the uniform mixed state over the subspace complementary to any upb is a bound entangled state we exhibit a tripartite 2 3 2 3 2 upb whose complementary mixed state has tripartite entanglement but no bipartite entanglement i e all three corresponding 2 3 4 bipartite mixed states are unentangled we show that members of a upb are not perfectly distinguishable by local positive operator valued measurements and classical communication s0031 9007 99 09360 6

convergence of an implicit finite element method for the landau lifshitz gilbert equation the landau lifshitz gilbert equation describes the dynamics of ferromagnetism where strong nonlinearity and nonconvexity are hard to tackle so far existing explicit schemes to approximate weak solutions suffer from severe time step restrictions in this paper we propose an implicit fully discrete scheme and verify unconditional convergence

geometric integration on spheres and some interesting applications geometric integration theory can be employed when numerically solving odes or pdes with constraints in this paper we present several one step algorithms of various orders for odes on a collection of spheres to demonstrate the versatility of these algorithms we present representative calculations for reduced free rigid body motion a conservative ode and a discretization of micromagnetics a dissipative pde we emphasize the role of isotropy in geometric integration and link numerical integration schemes to modern differential geometry through the use of partial connection forms this theoretical framework generalizes moving frames and connections on principal bundles to manifolds with nonfree actions

on the first integral and equivalence of nonlinear differential equations in this paper we study the equivalence of several different types of differential systems and provide a new method to find out the first integral of such differential systems we draw some sufficient conditions for a general time varying nonlinear differential system to be equivalent to a given differential system we use the obtained results to discuss the qualitative behavior of the periodic solutions of such time varying differential systems and get some interesting conclusions