exact evaluation of the collapse phase boundary for two dimensional directed polymers the phase boundary of the collapsed phase for a directed polymer on a square lattice in the presence of an attractive wall and monomer monomer interactions is calculated exactly using transfer matrix techniques the position of the multicritical point is also identified

an efficient multi step iterative method for computing the numerical solution of systems of nonlinear equations associated with odes an efficient multi step iterative method for computing the numerical solution of systems of nonlinear equations associated with odes

annihilating random walks in one dimensional disordered media we study diffusion limited pair annihilation a a to 0 on one dimensional lattices with inhomogeneous nearest neighbour hopping in the limit of infinite reaction rate we obtain a simple exact expression for the particle concentration rho k t of the many particle system in terms of the conditional probabilities p m t l 0 for a single random walker in a dual medium for some disordered systems with an initially randomly filled lattice this leads asymptotically to bar rho t bar p 0 2t 0 0 for the disorder averaged particle density we also obtain interesting exact relations for single particle conditional probabilities in random media related by duality such as random barrier and random trap systems for some specific random barrier systems the smoluchovsky approach to diffusion limited annihilation turns out to fail

non abelian symmetries of stochastic processes derivation of correlation functions for random vertex models and disordered interacting particle systems we consider systems of particles hopping stochastically on d dimensional lattices with space dependent probabilities we map the master equation onto an evolution equation in a fock space where the dynamics are given by a quantum hamiltonian continuous time or a transfer matrix discrete time using non abelian symmetries of these operators we derive duality relations expressing the time evolution of a given initial configuration in terms of correlation functions of simpler dual processes particularly simple results are obtained for the time evolution of the density profile as a special case we show that for any su 2 symmetric system the two point and three point density correlation functions in the n particle steady state can be computed from the probability distribution of a single particle moving in the same environment we apply our results to various models among them partial exclusion a simple diffusion reaction system and the two dimensional six vertex model with space dependent vertex weights for a random distribution of the vertex weights one obtains a version of the random barrier model describing diffusion of particles in disordered media we derive exact expressions for the averaged two point density correlation functions in the presence of weak correlated disorder

higher order multi step jarratt like method for solving systems of nonlinear equations this paper proposes a multi step iterative method for solving systems of nonlinear equations with a local convergence order of 3m4 where m 2 is the number of steps the multi step iterative method includes two parts the base method and the multi step part the base method involves two function evaluations two jacobian evaluations one lu decomposition of a jacobian and two matrixvector multiplications every stage of the multi step part involves the solution of two triangular linear systems and one matrixvector multiplication the computational efficiency of the new method is better than those of previously proposed methods the method is applied to several nonlinear problems resulting from discretizing nonlinear ordinary differential equations and nonlinear partial differential equations

basic representations of a2l 2 and dl 1 2 and the polynomial solutions to the reduced bkp hierarchies basic representations of a2l 2 and dl 1 2 are studied the weight vectors are represented in terms of schur s q functions the method to get the polynomial solutions to the reduced bkp hierarchies is shown to be equivalent to a certain rule in the maya game

some model theory of compact lie groups we consider questions of first order definability in a compact lie group g our main result is that if such g is simple and centerless then the lie group structure of g is first order definable from the abstract group structure along the way we also show i if g is non abelian and connected then a copy of the field 11t is interpretable in g and ii any 1 dimensional field interpretable in 11t is definably i e semialgebraically isomorphic to the ground field 11t

a multi parameter family of three step eighth order iterative methods locating a simple root a multi parameter family of three step eighth order iterative methods free from second derivatives are proposed in this paper to find a simple root of nonlinear algebraic equations convergence analysis as well as numerical experiments confirms the eighth order convergence and asymptotic error constants

an efficient jacobi pseudospectral approximation for nonlinear complex generalized zakharov system in this paper we derive an efficient spectral collocation algorithm to solve numerically the nonlinear complex generalized zakharov system gzs subject to initial boundary conditions the jacobi pseudospectral approximation is investigated for spatial approximation of the gzs it possesses the spectral accuracy in space the jacobi gauss lobatto quadrature rule is established to treat the boundary conditions and then the problem with its boundary conditions is reduced to a system of ordinary differential equations in time variable this scheme has the advantage of allowing us to obtain the spectral solution in terms of the jacobi parameters and which therefore means that the algorithm holds a number of collocation methods as special cases finally two illustrative examples are implemented to assess the efficiency and high accuracy of the jacobi pseudo spectral scheme

extinction survival and dynamical phase transition of branching annihilating random walk we analyze statistical properties of random walkers which disappear when they meet and make offsprings by a controllable rate numerical results for one two and three dimensions and for the sierpinski gasket are assessed in view of the mean field theory predictions universality classes are found to depend on the number of offsprings in space dimension less than 3

a new family of eighth order iterative methods for solving nonlinear equations a family of eighth order iterative methods with four evaluations for the solution of nonlinear equations is presented kung and traub conjectured that an iteration method without memory based on n evaluations could achieve optimal convergence order 2 n 1 the new family of eighth order methods agrees with the conjecture of kung traub for the case n 4 therefore this family of methods has efficiency index equal to 1 682 numerical comparisons are made with several other existing methods to show the performance of the presented methods

branching annihilating random walks in one dimension some exact results we derive a self duality relation for a one dimensional model of branching and annihilating random walkers with an even number of offspring on nearest neighbour sites with the duality relation and by deriving further exact results in some limiting cases involving fast diffusion we obtain new information on the location and nature of the phase transition line between an active stationary state non zero density and an absorbing state extinction of all particles thus clarifying some so far open problems in these limits the transition is mean field like but on the active side of the phase transition line the fluctuation in the number of particles deviates from its mean field value we also show that well within the active region of the phase diagram a finite system approaches the absorbing state very slowly on a time scale which diverges exponentially in system size in the absence of particle diffusion the branching process with infinite annihilation rate is strongly non ergodic

a six vertex model as a diffusion problem derivation of correlation functions a cellular automaton which describes diffusion of particles with exclusion in one dimension is shown to be equivalent to a six vertex model on a critical line the arrow arrow correlation function of the six vertex model is calculated exactly on this line using a transfer matrix method

a super fast tri parametric iterative method with memory for the first time a tri parametric method with memory for solving nonlinear equations is constructed at which a super fast convergence rate is achieved by using only three functional evaluations subsequently a very high computational efficiency index nearly 2 is attained finally to support the theory some numerical experiments are reported

a double optimal iterative algorithm in an affine krylov subspace for solving nonlinear algebraic equations for an n dimensional linear equations system liu 2014 has derived a double optimal solution in an affine m dimensional krylov subspace with m n an iterative algorithm based on the double optimal solution of the newton equation b u f in x u is proposed to solve a system of nonlinear algebraic equations f x 0 with dimension n by optimizing two merit functions u can be explicitly solved in the affine krylov subspace the resulting double optimal iterative algorithm doia is proven to be absolutely convergent with the square residual norm f 2 being reduced by b k u k 2 at each iteration and very time saving by merely inverting an m m positive definite matrix one time at each iterative step we can prove that such an algorithm leads to the largest convergence rate without needing to invert the n n matrix b some numerical examples are used to evaluate the performance of the doia where very fast convergence rates and saving the cpu time to find the solutions are observed

solving systems of nonlinear equations when the nonlinearity is expensive construction of multi step iterative method for solving system of nonlinear equations is considered when the nonlinearity is expensive the proposed method is divided into a base method and multi step part the convergence order of the base method is five and each step of multi step part adds additive factor of five in the convergence order of the base method the general formula of convergence order is 5 m 2 where m 3 is the step number for a single instance of the iterative method we only compute two jacobian and inversion of one jacobian is required the direct inversion of jacobian is avoided by computing lu factors the computed lu factors are used in the multi step part for solving five systems of linear equations that make the method computational efficient the distinctive feature of the underlying multi step iterative method is the single call to the computationally expensive nonlinear function and thus offers an increment of additive factor of five in the convergence order per single call the numerical simulations reveal that our proposed iterative method clearly shows better performance where the computational cost of the involved nonlinear function is higher than the computational cost for solving five lower and upper triangular systems

propagation and extinction in branching annihilating random walks we investigate the temporal evolution and spatial propagation of branching annihilating random walks in one dimension depending on the branching and annihilation rates a few particle initial state can evolve to a propagating finite density wave or extinction may occur in which the number of particles vanishes in the long time limit the number parity conserving case where 2 offspring are produced in each branching event can be solved exactly for unit reaction probability from which qualitative features of the transition between propagation and extinction as well as intriguing parity specific effects are elucidated an approximate analysis is developed to treat this transition for general baw processes a scaling description suggests that the critical exponents which describe the vanishing of the particle density at the transition are unrelated to those of conventional models such as reggeon field theory p a c s numbers 02 50 s 05 40 j 82 20 w

non solvable groups of morley rank 3 abstract we prove the following theorem let g be a connected centerless non solvable group of morley rank 3 whose proper definable subgroups have rank 1 then g has no involutions and for all x g cg x is connected our method permits us to deduce that son r is not stable for n 3

three step iterative methods with eighth order convergence for solving nonlinear equations a family of eighth order iterative methods for the solution of nonlinear equations is presented the new family of eighth order methods is based on king s fourth order methods and the family of sixth order iteration methods developed by chun et al per iteration the new methods require three evaluations of the function and one evaluation of its first derivative therefore this family of methods has the efficiency index which equals 1 682 kung and traub conjectured that a multipoint iteration without memory based on n evaluations could achieve optimal convergence order 2 n 1 thus we provide a new example which agrees with the conjecture of kung traub for n 4 numerical comparisons are made to show the performance of the presented methods

eighth order methods with high efficiency index for solving nonlinear equations in this paper we derive a new family of eighth order methods for solving simple roots of nonlinear equations by using weight function methods per iteration these methods require three evaluations of the function and one evaluation of its first derivative which implies that the efficiency indexes are 1 682 numerical comparisons are made to show the performance of the derived methods as shown in the illustration examples

field theory of branching and annihilating random walks we develop a systematic analytic approach to the problem of branching and annihilating random walks equivalent to the diffusion limited reaction processes 2a and a m 1 a where m 1 starting from the master equation a field theoretic representation of the problem is derived and fluctuation effects are taken into account via diagrammatic and renormalization group methods for d 2 the mean field rate equation which predicts an active phase as soon as the branching process is switched on applies qualitatively for both even and odd m but the behavior in lower dimensions is shown to be quite different for these two cases for even m and d near 2 the active phase still appears immediately but with nontrivial crossover exponents which we compute in an expansion in 2 d and with logarithmic corrections in d 2 however there exists a second critical dimension d c 4 3 below which a nontrivial inactive phase emerges with asymptotic behavior characteristic of the pure annihilation process this is confirmed by an exact calculation in d 1 the subsequent transition to the active phase which represents a new nontrivial dynamic universality class is then investigated within a truncated loop expansion which appears to give a correct qualitative picture the model with m 2 is also generalized to n species of particles which provides yet another universality class and which is exactly solvable in the limit n for odd m we show that the fluctuations of the annihilation process are strong enough to create a nontrivial inactive phase for all d 2 in this case the transition to the active phase is in the directed percolation universality class finally we study the modification when the annihilation reaction is 3a when m 0 mod 3 the system is always in its active phase but with logarithmic crossover corrections for d 1 while the other cases should exhibit a directed percolation transition out of a fluctuation driven inactive phase