random walk in chemical space of cantor dust as a paradigm of superdiffusion we point out that the chemical space of a totally disconnected cantor dust k n ensuremath subset e n is a compact metric space c n with the spectral dimension d s d ensuremath ell ngd where d and d ensuremath ell n are the fractal and chemical dimensions of k n respectively hence we can define a random walk in the chemical space as a markovian gaussian process the mapping of a random walk in c n into k n ensuremath subset e n defines the quenched l evy flight on the cantor dust with a single step duration independent of the step length the equations describing the superdiffusion and diffusion reaction front propagation ruled by the local quenched l evy flight on k n ensuremath subset e n are derived the use of these equations to model superdiffusive phenomena observed in some physical systems in which propagators decay faster than algebraically is discussed

inertia controlling methods for general quadratic programming active set quadratic programming qp methods use a working set to define the search direction and multiplier estimates in the method proposed by fletcher in 1971 and in several subsequent mathematically equivalent methods the working set is chosen to control the inertia of the reduced hessian which is never permitted to have more than one nonpositive eigenvalue such methods will be called inertia controlling this paper presents an overview of a generic inertia controlling qp method including the equations satisfied by the search direction when the reduced hessian is positive definite singular and indefinite recurrence relations are derived that define the search direction and lagrange multiplier vector through equations related to the karush kuhn tucker system discussion is included of connections with inertia controlling methods that maintain an explicit factorization of the reduced hessian matrix

survival probability and order statistics of diffusion on disordered media we investigate the first passage time t j n to a given chemical or euclidean distance of the first j of a set of n ensuremath gg 1 independent random walkers all initially placed on a site of a disordered medium to solve this order statistics problem we assume that for short times the survival probability the probability that a single random walker is not absorbed by a hyperspherical surface during some time interval decays for disordered media in the same way as for euclidean and some class of deterministic fractal lattices this conjecture is checked by simulation on the incipient percolation aggregate embedded in two dimensions arbitrary moments of t j n are expressed in terms of an asymptotic series in powers of 1 mathrm ln n which is formally identical to those found for euclidean and some class of deterministic fractal lattices the agreement of the asymptotic expressions with simulation results for the two dimensional percolation aggregate is good when the boundary is defined in terms of the chemical distance the agreement worsens slightly when the euclidean distance is used

on practical conditions for the existence and uniqueness of solutions to the general equality quadratic programming problem we present practical conditions under which the existence and uniqueness of a finite solution to a given equality quadratic program may be examined different sets of conditions allow this examination to take place when null space range space or lagrangian methods are used to find stationary points for the quadratic program

on the convegence of a sequential penalty function method for constrained minimization the convergence behaviour of a class of iterative methods for solving the constrained minimization problem is analysed the methods are based on the sequential minimization of a simple differentiable penalty function they are sufficiently general to ensure global convergence of the iterates to the solution of the problem at an asymptotic two step q superlinear rate

what are schur complements anyway abstract given a matrix m a b c d the schur complements of a in m are the matrices of the form s d cab where a is a generalized inverse of a we survey several recent characterizations of schur complements and discuss where they arose and how they are related

new crash procedures for large systems of linear constraints many algorithms for solving linearly constrained optimization problems maintain sets of basic variables the calculation of the initial basis is of great importance as it determines to a large extent the amount of computation that will then be required to solve the problem in this paper we suggest a number of simple methods for obtaining an initial basis and perform tests to indicate how they perform on a variety of real life problems

newton methods for large scale linear inequality constrained minimization newton methods of the linesearch type for large scale minimization subject to linear inequality constraints are discussed the purpose of the paper is twofold i to give an active set type method with the ability to delete multiple constraints simultaneously and ii to give a relatively short general convergence proof for such a method it is also discussed how multiple constraints can be added simultaneously the approach is an extension of a previous work by the same authors for equality constrained problems it is shown how the search directions can be computed without the need to compute the reduced hessian of the objective function the convergence analysis states that every limit point of a sequence of iterates satisfies the second order necessary optimality conditions

the similarity group and anomalous diffusion equations a number of distinct differential equations known as generalized diffusion equations have been proposed to describe the phenomenon of anomalous diffusion on fractal objects although all are constructed to correctly reproduce the basic subdiffusive property of this phenomenon using similarity methods it becomes very clear that this is far from sufficient to confirm their validity the similarity group that they all have in common is the natural basis for making comparisons between these otherwise different equations and a practical basis for comparisons between the very different modelling assumptions that their solutions each represent similarity induces a natural space in which to compare these solutions both with one another and with data from numerical experiments on fractals it also reduces the differential equations to extra ordinary ones which are presented here for the first time it becomes clear here from this approach that the proposed equations cannot agree even qualitatively with either each other or the data suggesting that a new approach is needed

diffusion equations for a markovian jumping process we consider a markovian jumping process which is defined in terms of the jump size distribution and the waiting time distribution with a position dependent frequency in the diffusion limit we assume the power law form for the frequency for small steps we derive the fokker planck equation and show the presence of the normal diffusion subdiffusion and superdiffusion for the l evy distribution of the step size we construct a fractional equation which possesses a variable coefficient and solve it in the diffusion limit then we calculate fractional moments and define the fractional diffusion coefficient as a natural extension to the cases with the divergent variance we also solve the master equation numerically and demonstrate that there are deviations from the l evy stable distribution for large wave numbers

convergence properties of trust region methods for linear and convex constraints we develop a convergence theory for convex and linearly constrained trust region methods which only requires that the step between iterates produce a sufficient reduction in the trust region subproblem global convergence is established for general convex constraints while the local analysis is for linearly constrained problems the main local result establishes that if the sequence converges to a nondegenerate stationary point then the active constraints at the solution are identified in a finite number of iterations as a consequence of the identification properties we develop rate of convergence results by assuming that the step is a truncated newton method our development is mainly geometrical this approach allows the development of a convergence theory without any linear independence assumptions

newton s method for large bound constrained optimization problems we analyze a trust region version of newton s method for bound constrained problems our approach relies on the geometry of the feasible set not on the particular representation in terms of constraints the convergence theory holds for linearly constrained problems and yields global and superlinear convergence without assuming either strict complementarity or linear independence of the active constraints we also show that the convergence theory leads to an efficient implementation for large bound constrained problems

on the solution of highly degenerate linear programs the solution of scheduling problems often gives rise to highly degenerate linear programmes which cause significant computational difficulties for the revised simplex method wolfe s highly effective ad hoc method for overcoming the cycling or stalling problems associated with degeneracy is described here it is given a geometric interpretation in terms of finding a direction of recession for a reduced problem which is fundamental to a full understanding of the procedure an example of an aircrew scheduling problem is used to illustrate the effectiveness of the method

zero constant formula for first passage observables in bounded domains in this letter we develop an analytical approach which provides an explicit determination of mean first passage times mfpts for random walks in bounded domains for a wide class of transport processes in particular we derive for the first time explicit expressions of mfpts for emblematic models of transport in complex media such as diffusion on deterministic and random fractals this approach relies on a scale invariance hypothesis and a large volume expansion of the mfpt which actually proves to be very accurate even for small system sizes as shown by numerical simulations this explicit determination of mfpts can be straightforwardly generalized to further useful first passage observables such as occupation times and splitting probabilities

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

an active set quasi newton method with projected search for bound constrained minimization we analyze an active set quasi newton method for large scale bound constrained problems our approach combines the accurate active set identification function and the projected search both of these strategies permit fast change in the working set the limited memory method is employed to update the inactive variables while the active variables are updated by simple rules a further division of the active set enables the global convergence of the new algorithm numerical tests demonstrate the efficiency and performance of the present strategy and its comparison with some existing active set strategies

a practical anti cycling procedure for linearly constrained optimization a procedure is described for preventing cycling in active set methods for linearly constrained optimization including the simplex method the key ideas are a limited acceptance of infeasibilities in all variables and maintenance of a working feasibility tolerance that increases over a long sequence of iterations the additional work per iteration is nominal and stalling cannot occur with exact arithmetic the method appears to be reliable based on computational results for the first 53 linear programming problems in thenetlib set

a new symmetric five diagonal finite difference method for computing eigenvalues of fourth order two point boundary value problems abstract we present a new symmetric five diagonal finite difference method for computing eigenvalues of two point boundary value problems involving a fourth order differential equation

short time regime propagator in fractals the propagator p r t for fractals in the short time regime i e the probability of finding at distance r at time t a particle that diffuses in a fractal substrate when ensuremath xi r sqrt 2d t 1 d w ensuremath gg 1 is studied in order to elucidate its full functional form for finitely ramified fractals it is shown and for any other self similar media conjectured that the short time propagator is given by p r t ensuremath approx p 0 t ensuremath d s 2 ensuremath xi ensuremath alpha mathrm exp ensuremath c ensuremath xi ensuremath nu where ensuremath nu d w d w ensuremath 1 and ensuremath alpha ensuremath nu 2 ensuremath d f d f and d w being the fractal and random walk dimension of the medium respectively the value for ensuremath nu agrees with that generally accepted however our result for the as yet not well established value of ensuremath alpha differs from other recent proposals we have checked these various short time propagator proposals by comparing them to the short time propagator calculated numerically for the sierpinsky gasket our numerical results are precise enough to clearly support the validity of the short time propagator proposed here in particular the validity of our relation for ensuremath alpha and to rule out the others

equivalence of some quadratic programming algorithms we formulate a general algorithm for the solution of a convex but not strictly convex quadratic programming problem conditions are given under which the iterates of the algorithm are uniquely determined the quadratic programming algorithms of fletcher gill and murray best and ritter and van de panne and whinston dantzig are shown to be special cases and consequently are equivalent in the sense that they construct identical sequences of points the various methods are shown to differ only in the manner in which they solve the linear equations expressing the kuhn tucker system for the associated equality constrained subproblems equivalence results have been established by goldfarb and djang for the positive definite hessian case our analysis extends these results to the positive semi definite case

convergence to second order stationary points in inequality constrained optimization we propose a new algorithm for the nonlinear inequality constrained minimization problem and prove that it generates a sequence converging to points satisfying the kkt second order necessary conditions for optimality the algorithm is a line search algorithm using directions of negative curvature and it can be viewed as a nontrivial extension of corresponding known techniques from unconstrained to constrained problems the main tools employed in the definition and in the analysis of the algorithm are a differentiable exact penalty function and results from the theory of lc1 functions

a five diagonal finite difference method based on mixed type interpolation for computing eigenvalues of fourth order two point boundary value problems a modified finite difference method based on mixed type interpolation is derived and applied for computing eigenvalues of fourth order two point boundary value problems a parallelized version of the resulting algorithm is discussed and implemented on a helios based parallel processing transputer system some numerical experiments are performed

fractional diffusion equation and green function approach exact solutions abstract we investigate the solutions of a fractional diffusion equation with radial symmetry by using the green function approach and by taking the n dimensional case into account in our analysis a spatial time dependent diffusion coefficient is considered i e d r t d t 1 r the presence of external forces f r k r e with e 1 and f r kr k r e is also taken into account in particular we discuss the results obtained by employing boundary conditions defined on a finite interval and afterwards the analysis is extended to a semi infinite interval finally we also discuss a rich class of diffusive processes that can be obtained from the results presented in this work

use of a shooting method to compute eigenvalues of fourth order two point boundary value problems abstract the shooting method for solving regular two point boundary value problems tpbvps can also be used to solve eigenvalue problems of that type the method in the past has been used to solve eigenvalue problems related to partial differential equations as in the helmholtz equation for example the present new application to computing eigenvalues related to fourth order ordinary differential equations is shown to be quite straightforward and solutions are obtained in a few seconds on modern workstations

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

first passage properties of mortal random walks ballistic behavior effective reduction of dimensionality and scaling functions for hierarchical graphs we consider a mortal random walker on a family of hierarchical graphs in the presence of some trap sites the configuration comprising the graph the starting point of the walk and the locations of the trap sites is taken to be exactly self similar as one goes from one generation of the family to the next under these circumstances the total probability that the walker hits a trap is determined exactly as a function of the single step survival probability q of the mortal walker on the nth generation graph of the family this probability is shown to be given by the nth iterate of a certain scaling function or map q f q the properties of the map then determine in each case the behavior of the trapping probability the mean time to trapping the temporal scaling factor governing the random walk dimension on the graph and other related properties the formalism is illustrated for the cases of a linear hierarchical lattice and the sierpinski graphs in two and three euclidean dimensions we find an effective reduction of the random walk dimensionality due to the ballistic behavior of the surviving particles induced by the mortality constraint the relevance of this finding for experiments involving travel times of particles in diffusion decay systems is discussed