when is the 2 2 matrix ring over a commutative local ring strongly clean a ring r with identity is called strongly clean if every element of r is the sum of an idempotent and a unit that commute local rings are strongly clean it is unknown when a matrix ring is strongly clean however it is known from j chen x yang y zhou on strongly clean matrix and triangular matrix rings preprint 2005 that for any prime number p the 2 2 matrix ring m2 z p is strongly clean where z p is the ring of p adic integers but m2 z p is not strongly clean where z p is the localization of z at the prime ideal generated by p let r be a commutative local ring a criterion in terms of solvability of a simple quadratic equation in r is obtained for m2 r to be strongly clean as consequences m2 r is strongly clean iff m2 r x is strongly clean iff m2 r x xn is strongly clean iff m2 rc2 is strongly clean

dynamics of breathers in discrete nonlinear schro dinger models abstract we review some recent results concerning the existence and stability of spatially localized and temporally quasiperiodic non stationary excitations in discrete nonlinear schrodinger dnls models in two dimensions we show the existence of linearly stable stationary and non stationary localized vortex like solutions we also show that stationary on site localized excitations can have internal breathing modes which are spatially localized and symmetric the excitation of these modes leads to slowly decaying quasiperiodic oscillations finally we show that for some generalizations of the dnls equation where bistability occurs a controlled switching between stable states is possible by exciting an internal breathing mode above a threshold value

discrete solitons in nonlinear schrodinger lattices with a power law nonlinearity abstract we study the discrete nonlinear schrodinger lattice model with the onsite nonlinearity of the general form u 2 u we systematically verify the conditions for the existence and stability of discrete solitons in the one dimensional version of the model predicted by means of the variational approximation va and demonstrate the following monostability of fundamental solitons fss in the case of the weak nonlinearity 2 1 3 68 bistability in a finite range of values of the soliton s power for 3 68 2 1 5 and the presence of a threshold minimum norm of the fs for 2 1 5 we also perform systematic numerical simulations to study higher order solitons in the same general model i e bound states of the fss while all in phase bound states are unstable stability regions are identified for antisymmetric double solitons and their triple counterparts these numerical findings are supplemented by an analytical treatment of the stability problem which allows quantitively accurate predictions for the stability features of such multipulses when these waveforms are found to be unstable we show by means of direct simulations that they self trap into a persistent lattice breather or relax into a stable fs or sometimes decay completely

diagonability of idempotent matrices over noncommutative rings abstract let r be an arbitrary ring in this paper the following statements are proved a each idempotent matrix over r can be diagonalized if and only if each idempotent matrix over r has a characteristic vector b an idempotent matrix over r can be diagonalized under a similarity transformation if and only if it is equivalent to a diagonal matrix a and b generalize foster s and steger s theorems to arbitrary rings we give some new results about 0 similarity of idempotent matrices over r

quasi periodic solutions of nonlinear schrodinger equations of higher dimension abstract it is shown that there are plenty of quasi periodic solutions of nonlinear schrodinger equations of higher spatial dimension where the dimension of the frequency vectors of the quasi periodic solutions are equal to that of the space

hamiltonian hopf bifurcations in the discrete nonlinear schrodinger trimer oscillatory instabilities quasi periodic solutions and a new type of self trapping transition oscillatory instabilities in hamiltonian anharmonic lattices are known to appear through hamiltonian hopf bifurcations of certain time periodic solutions of multibreather type here we analyse the basic mechanisms for this scenario by considering the simplest possible model system of this kind where they appear the three site discrete nonlinear schrodinger model with periodic boundary conditions the stationary solution having equal amplitude and opposite phases on two sites and zero amplitude on the third is known to be unstable for an interval of intermediate amplitudes we numerically analyse the nature of the two bifurcations leading to this instability and find them to be of two different types close to the lower amplitude threshold stable two frequency quasi periodic solutions exist surrounding the unstable stationary solution and the dynamics remains trapped around the latter so that in particular the amplitude of the originally unexcited site remains small by contrast close to the higher amplitude threshold all two frequency quasi periodic solutions are detached from the unstable stationary solution and the resulting dynamics is of population inversion type involving also the originally unexcited site

internal modes of envelope solitons abstract we extend the concept of internal mode to envelope solitons and show that this mode is responsible for long lived weakly damped periodic oscillations of the soliton amplitude observed in numerical simulations we present analytical and numerical results for solitons of the generalized nonlinear schrodinger equation and analyze the example of the cubic quintic nonlinearity in more detail we obtain also analytical criteria for the existence and bifurcations of the soliton internal mode and calculate the rate of the radiation induced damping of the soliton oscillations induced by excitation of the internal mode

strongly clean matrix rings over local rings an element of a ring r with identity is called strongly clean if it is the sum of an idempotent and a unit that commute and r is called strongly clean if every element of r is strongly clean in this paper we determine when a 2 2 matrix a over a commutative local ring is strongly clean several equivalent criteria are given for such a matrix to be strongly clean consequently we obtain several equivalent conditions for the 2 2 matrix ring over a commutative local ring to be strongly clean extending a result of chen yang and zhou

strong cleanness of generalized matrix rings over a local ring abstract let k s r be the generalized matrix ring over a ring r with multiplier s for a general local ring r and a central element s in the jacobson radical of r necessary and sufficient conditions are obtained for k s r to be a strongly clean ring for a commutative local ring r and an arbitrary element s in r criteria are obtained for a single element of k s r to be strongly clean and respectively for the ring k s r to be strongly clean specializing to s 1 yields some known results new families of strongly clean rings are presented

monogenic hopf orders and associated orders of valuation rings abstract let r be a complete discrete valuation ring of mixed characteristic with perfect residue field and let h be a finite local commutative r hopf algebra we consider when there exists a finite extension of the field of fractions of r whose valuation ring is a galois h object if this occurs then h is monogenic conversely if h is also cocommutative and h is monogenic then there exists a valuation ring which is a galois h object to prove this result we represent h as the kernel of an isogeny of a special type between formal groups over r we deduce that if a is a finite abelian r hopf algebra such that both a and its dual are local then a is the associated order of a valuation ring if and only if the dual of a is monogenic

bound states of two dimensional solitons in the discrete nonlinear schrodinger equation the existence and stability of bound states bss of solitary excitations in the two dimensional discrete nonlinear schrodinger lattice are considered stable bss of two or three solitons are categorized according to their vorticity s s 0 corresponds to ordinary pulses while s 1 corresponds to discrete vortex solitons interactions of s 0 solitons are found to have clear particle like characteristics which can be very accurately predicted in the framework of the perturbation theory that we develop triangular bound states of three solitons are also constructed and particle like characteristics in their behaviour are followed in numerical experiments finally stable bound states of two vortex solitons and s 0 1 stable complexes are constructed for the latter state we find a weak dependence of the interaction generated stability eigenvalue on the distance between the two solitons

construction of quasi periodic breathers via kam technique by developing a kam theorem which involves an infinitely multiple normal frequency it is shown that there are plenty of breathers quasi periodic in time and super exponentially localized in space for the networks of weakly coupled oscillators this answers an open problem by aubry a2 in case the linearized system has no continuous spectrum

on localization in the discrete nonlinear schro dinger equation abstract for some values of the grid resolution depending on the nonlinearity the discrete nonlinear schrodinger equation with arbitrary power nonlinearity can be approximated by the corresponding continuum version of the equation when the discretization becomes too coarse the discrete equation exhibits localization in regimes where blow up cannot occur in the continuum system this phenomenon is investigated numerically and the grid resolution at which the transition occurs is determined

strongly clean triangular matrix rings over local rings abstract an element of a ring is called strongly clean if it can be written as the sum of a unit and an idempotent that commute a ring is called strongly clean if each of its elements is strongly clean in this paper we investigate conditions on a local ring r that imply that t n r is a strongly clean ring it is shown that this is the case for commutative local rings r as well as for a host of other classes of local rings an example of a local ring a for which t 2 a is not strongly clean is also given

breathers in nonlinear lattices existence linear stability and quantization abstract it has been proved by the principle of anti integrable limit renamed here anti continuous limit and the implicit function theorem that under rather weak hypotheses there exist spatially localized and time periodic solutions breathers in arrays of nonlinear coupled classical oscillators provided the coupling is not too strong the models can be translationally invariant or not and be of any dimension there are also time periodic solutions corresponding to arbitrary distributions on the lattice of such solutions multibreathers the whole set of these solutions can be labeled by a discrete coding sequence the condition that the frequency of the breather or multibreather solution has no harmonics in the phonon spectrum can be discarded for a large subset of spatially extended time periodic solutions called phonobreather which correspond to special codes which can be also spatially chaotic and where no oscillator is at rest these results extend almost without change to models with coupled rotors the existence of rotobreathers which are exact solutions consisting of a single rotor rotating while the other rotors oscillate is proved there are also multirotobreather solutions and phono rotobreathers etc it is also shown that the phase of the single breathers constituting a multibreather or multirotobreather solution can be submitted to a phase torsion these new dynamical structures carry a nonzero energy flow in models with two dimensions and more the phase torsion of multibreather solutions can generate new dynamical solutions with vortices in the energy flow the dynamics of a quantum electron coupled to a classical lattice is also studied within the same approach from the anti continuous limit the existence of two kinds of polarobreathers which are spatially exponentially localized dynamical solutions is proved for a small enough electronic transfer integral the polarobreathers of the first kind include the standard static polaron the lattice configuration is static time independent when the electron is in an excited state i e the electronic oscillator oscillates periodically they can be spatially chaotic a polarobreather of the second kind corresponds to a localized periodic lattice oscillation associated with a localized quasiperiodic electronic oscillation the most simple of these solutions are those for which both the lattice oscillation and the electron wave function are mostly concentrated close to the same single site the standard floquet analysis of the linear stability of the breather and multibreather solutions is shown to be related with the band spectrum property of the newton operator involved in the implicit function theorem the krein theory of bifurcations is reinterpreted from this point of view the linear stability of the single breather and rotobreather solutions is then proved at least at small enough coupling finally it is shown that in the quantum version of translationally invariant classical models where there cannot exist in principle any strictly localized excitation there are narrow bands of excitations consisting of bound states of many bosons which are the quantum counterpart of the classical breathers the bandwidth of these excitations goes exponentially to zero with the number of bound phonons this theory provides simultaneously existence theorems and a new powerful and accurate numerical method for calculating practically any of the solutions which has been formally predicted to exist these numerical applications are currently written elsewhere

existence and stability of 3 site breathers in a triangular lattice we find conditions for existence and stability of various types of discrete breather concentrated around three central sites in a triangular lattice of one dimensional hamiltonian oscillators with on site potential and nearest neighbour coupling in particular we confirm that it can support non reversible breather solutions despite the time reversible character of the system they carry a net energy flux and can be called vortex breathers we prove that there are parameter regions for which they are linearly stable for example in a lattice consisting of coupled morse oscillators whereas the related reversible breathers are unstable thus non reversible breathers can be physically relevant

strong cleanness of the 2 2 matrix ring over a general local ring abstract a ring r is called strongly clean if every element of r is the sum of a unit and an idempotent that commute with each other a recent result of borooah diesl and dorsey g borooah a j diesl t j dorsey strongly clean matrix rings over commutative local rings j pure appl algebra 212 1 2008 281 296 completely characterized the commutative local rings r for which m n r is strongly clean for a general local ring r and n 1 however it is unknown when the matrix ring m n r is strongly clean here we completely characterize the local rings r for which m 2 r is strongly clean

persistence and stability of discrete vortices in nonlinear schrodinger lattices abstract we study discrete vortices in the two dimensional nonlinear schrodinger lattice with small coupling between lattice nodes the discrete vortices in the anti continuum limit of zero coupling represent a finite set of excited nodes on a closed discrete contour with a non zero charge using the lyapunov schmidt reductions we analyze continuation and termination of the discrete vortices for small coupling between lattice nodes an example of a square discrete contour is considered that includes the vortex cell also known as the off site vortex we classify families of symmetric and asymmetric discrete vortices that bifurcate from the anti continuum limit we predict analytically and confirm numerically the number of unstable eigenvalues associated with each family of such discrete vortices