hamiltonian connectivity of the wk recursive network with faulty nodes a graph g is said to be hamiltonian connected if there is a hamiltonian path between every two distinct nodes of g let f denote the set of faulty nodes of g then g is f node hamiltonian connected if g f is hamiltonian connected we use k d t to denote a wk recursive network of level t each of whose basic modules is a d node complete graph compared with other networks it is rather difficult to construct a hamiltonian path between two arbitrary nodes in a faulty wk recursive network in this paper we show that k d t is d 4 node hamiltonian connected since the connectivity of k d t is d 1 the result is optimal in the worst case

approximating the longest paths in grid graphs abstract in this paper we consider the problem of approximating the longest path in a grid graph that possesses a square free 2 factor by 1 analyzing several characteristics of four types of cross cells and 2 presenting three cycle merging operations and one extension operation we propose an algorithm that finds in an n vertex grid graph g a long path of length at least 5 6 n 2 our approximation algorithm runs in quadratic time in addition to its theoretical value our work has also an interesting application in image guided maze generation

the analytic solutions of some boundary layer problems in the theory of brownian motion the authors give analytic solutions of the generalised albedo and milne problems for the uhlenbeck ornstein process and discuss the relation between these and the original wang uhlenbeck 1945 problem of determining the distribution of first passage times

the density profile for the klein kramers equation near an absorbing wall we derive asymptotic series for the expansion coefficients of a function in terms of the pagani functions which occur in the boundary layer solutions of the klein kramers equation the results enable us to determine the density profile in the stationary solution of this equation near an absorbing wall from the numerically determined velocity distribution at the wall with an accuracy of about 2 we also obtain information about the analytic behavior of the density profile this profile increases near the wall with the square root of the distance to the wall finally the asymptotic analysis leads to an understanding of the slow convergence of variational approximations to the solution of the absorbing wall problem and of the exponents that occur when one studies the variational approximations to various quantities of interest as functions of the number of terms in the variational ansatz this is used to obtain a better variational estimate for the density at the wall

hamiltonian paths in l shaped grid graphs grid graphs are subgraphs of the infinite 2 dimensional integer grid the hamiltonian path problem for general grid graphs is a well known np complete problem in this paper we present necessary and sufficient conditions for the existence of a hamiltonian path between two given vertices in l shaped grid graphs we also show that a hamiltonian path between two given vertices of a l shaped grid graph can be computed in linear time the hamiltonian path problem is a well known np complete problem we give necessary and sufficient conditions for the existence of a hamiltonian s t path in l shaped grid graphs we show that a hamiltonian s t path in l shaped grid graph can be computed in linear time if it does exist

the analytic structure of the stationary kinetic boundary layer for brownian particles near an absorbing wall a method for solving kinetic boundary layer problems for the 1d klein kramers equation the kinetic equation for brownian particles was proposed a few years ago by marshall and watson 1987 for the special case of particles moving under the influence of a constant or zero external force in this paper the authors apply this method to a number of classical stationary boundary layer problems for completely or partially absorbing walls these are the milne problem in which a current flows towards the wall from infinity and the albedo problem in which particles are injected into the system at the wall with a prescribed velocity distribution the solutions are known to be non analytic at the wall for zero normal velocity the authors pay particular attention to the nature of this singularity for several special cases the results are compared with results obtained by approximate methods

the riemann hilbert problem for nonsymmetric systems a comparison of the riemann hilbert problem and the wiener hopf factorization problem arising in the solution of half space singular integral equations is presented emphasis is on the factorization of functions lacking the reflection symmetry usual in transport theory

exact solution of the reaction diffusion problem for a particle generating band on a surface by riemann hilbert matching the existence of a class of exact solutions for diffusion reaction problems is demonstrated for the example of a band the method may be applied to any particle source which can be analytically mapped to a band or a disc situated on an infinite plane a simple asymptotic expression for the experimentally measurable collector efficiency current current response function has been found

a three dimensional singularly perturbed conservative system with symmetry breaking in this paper we present an analysis of a three dimensional singularly perturbed conservative system we add a constant vector in the vector field to remove one of the symmetries in the system using the geometric argument and a theorem which is derived from the implicit function theorem we prove the existence of equilibria in the system and also derive some local bifurcations of these equilibria i e saddle node bifurcations we also show that although we have two saddle nodes in the system the codimension 2 bifurcation called the cusp bifurcation is not possible more sophisticated bifurcations such as the hopf bifurcation and the bifurcation of the created periodic solution are derived by using the numerical continuation software matcont following the periodic solution we found a sequence of period doubling bifurcations which leads to the existence of infinitely many periodic solutions the coexistence of two attractors is an interesting phenomenon which is observed in this paper using asymptotic analysis we discuss the dynamics in the neighborhood of a particular line in phase space at which the competition between these attractors takes place

the growth of small droplets from a gas mixture results and estimates for very small radii abstract to test and to complement the moment method for solving kinetic equations in the space outside of a sphere that absorbs one component out of a gas mixture we develop a solution method especially suitable for radii smaller or not much larger than a mean free path the method is used to calculate the stationary current towards an absorbing sphere suspended in a gas mixture that is in equilibrium far from the sphere for the simplest kinetic equation a variant of the linear bgk equation and for radii below two mean free paths this procedure gives the current towards the sphere to within an accuracy of at most a few tenths of a percent for radii around two mean free paths the new results agree with those of a variant of the moment method for lower radii discrepancies of up to a few percent are found our new method does not work for the klein kramers equation a kinetic equation for brownian particles for the latter equation we obtain an estimate for the current at very low radii from the free space solution by uhlenbeck and ornstein its numerical evaluation strongly indicates an analytic behavior for small radii different from the one found in the bgk case

first passage times for the uhlenbeck ornstein process a numerical solution is obtained for the laplace transformed backward kramers equation from which the mean first passage time may be obtained the main difficulties are associated with a the parabolic nature of the time development operator and b the existence of a double structure in the solution near the absorbing barrier both of these difficulties are resolved by computational methods derived from boundary layer theory the reliability of the method is assessed by comparing its results with an earlier analytic solution for the case of a uniform force field the authors also present the results for a harmonic force field for which no analytic solution is yet known

hamiltonian connectivity and strongly hamiltonian laceability of folded hypercubes in this paper we analyze a hypercube like structure called the folded hypercube which is basically a standard hypercube with some extra links established between its nodes we first show that the n dimensional folded hypercube is bipartite when n is odd we also show that the n dimensional folded hypercube is strongly hamiltonian laceable when n is odd and is hamiltonian connected when n 1 or n 2 is even

the kinetic boundary layer for the klein kramers equation a new numerical approach we explore a numerical technique for determining the structure of the kinetic boundary layer of the klein kramers equation for noninteracting brownian particles in a fluid near a wall that absorbs the brownian particles the equation is of interest in the theory of diffusion controlled reactions and of the coagulation of colloidal suspensions by numerical simulation of the langevin equation equivalent to the klein kramers equation we amass statistics of the velocities at the first return to the wall and of the return times for particles injected into the fluid at the wall with given velocities the data can be used to construct the solutions of the standard problems at an absorbing wall the milne and the albedo problem we confirm and extend earlier results by burschka and titulaer obtained by a variational method vexed by the slow convergence of the underlying eigenfunction expansion we briefly discuss some further boundary layer problems that can be attacked by exploiting the results reported here

half range completeness for the fokker planck equation half range completeness theorems are proved for eigenfunctions associated to the one dimensional fokker planck equation in a semi infinite medium existence and uniqueness results for perfectly absorbing partially absorbing and purely specularly reflecting boundary conditions are deduced for the stationary and time dependent problems similar results are obtained for a slab geometry

the kinetic boundary layer for the fokker planck equation with absorbing boundary the fokker planck equation for the distribution of position and velocity of a brownian particle is a particularly simple linear transport equation its normal solutions and an apparently complete set of stationary boundary layer solutions can be determined explicitly by a numerical algorithm we select linear combinations of them that approximately fulfill the boundary condition for a completely absorbing plane wall and that approach a linearly increasing position space density far from the wall various aspects of these approximate solutions are discussed in particular we find that the extrapolated asymptotic density reaches zero at a distance xm beyond the wall we find xm 1 46 in units of the velocity persistence length of the brownian particle this study was motivated by certain problems in the theory of diffusion controlled reactions and the results might be used to test approximate theories employed in that field

non resonant double hopf bifurcations the complex case the analysis of the unfolding of a non resonant double hopf singularity is considered using the frequency domain and the normal form methodologies a higher order harmonic balance and the evaluation of floquet multipliers are used to obtain accurate approximations of the limit cycles and its local bifurcations this type of hybrid methodology using harmonic balance and normal forms gives a complementary view in the vicinity of the singularity more specifically some of the bifurcation curves arising in the unfolding of the double hopf bifurcation are computed with great accuracy using the harmonic balance method compared to the classical normal form however this last method is able to predict the appearance of complex quasiperiodic behavior such as three dimensional 3d torus numerical simulations corroborate this observation

local bifurcations and a survey of bounded quadratic systems abstract this paper presents a survey of the known results for bounded quadratic systems as well as a study of the local bifurcations that occur at critical points of such systems it is shown that the only finite codimension bifurcations that occur at a critical point of a bounded quadratic system are the saddle node and the hopf takens bifurcations of codimensions 1 and 2 and the bogdanov takens bifurcations of codimensions 2 and 3 furthermore it is shown that whenever a bounded quadratic system has one of these critical points then a full generic unfolding of the critical point exists in the class of bounded quadratic systems finally we give a complete list of those limit periodic sets whose finite cyclicity still needs to be established in order to obtain the existence of a finite upper bound for the number of limit cycles that can occur in a bounded quadratic system

chaos and strange attractors in coupled oscillators with energy preserving nonlinearity we present an analysis for a particular singularly perturbed conservative system this system comes from the normal form of two coupled oscillator systems with widely separated frequencies and energy preserving nonlinearity the analysis is done in this paper for a degenerate case of such a system while the generic one has been treated in the literature to understand the relation with the strong resonance case we have computed the normal form of the 2 1 resonance and found that the latter is contained in our system we present a theorem that gives the existence of a nontrivial equilibrium for a general singularly perturbed conservative system we detect that the nontrivial equilibrium undergoes two hopf bifurcations furthermore the periodic solutions created through these hopf bifurcations undergo a sequence of period doubling bifurcations this leads to the presence of chaotic dynamics through shil nikov bifurcation of a homoclinic orbit also we measure the size of the chaotic attractor which is created in our system

hamilton paths in grid graphs a grid graph is a node induced finite subgraph of the infinite grid it is rectangular if its set of nodes is the product of two intervals given a rectangular grid graph and two of its nodes we give necessary and sufficient conditions for the graph to have a hamilton path between these two nodes in contrast the hamilton path and circuit problem for general grid graphs is shown to be np complete this provides a new relatively simple proof of the result that the euclidean traveling salesman problem is np complete

widely separated frequencies in coupled oscillators with energy preserving quadratic nonlinearity in this paper we present an analysis of a system of coupled oscillators suggested by atmospheric dynamics we make two assumptions for our system the first assumption is that the frequencies of the characteristic oscillations are widely separated and the second is that the nonlinear part of the vector field preserves the distance to the origin using the first assumption we prove that the reduced normal form of our system has an invariant manifold which exists for all values of the parameters this invariant manifold cannot be perturbed away by including higher order terms in the normal form using the second assumption we view the normal form as an energy preserving three dimensional system which is linearly perturbed restricting ourselves to a small perturbation the flow of the energy preserving system is used to study the flow in general we present a complete study of the flow of the energy preserving system and its bifurcations using these results we return to the dissipative system and provide the condition for having a hopf bifurcation of one of the two equilibria of the perturbed system we also numerically follow the periodic solution created via the hopf bifurcation and find a sequence of period doubling and fold bifurcations also a torus or neimark sacker bifurcation

a drop of ink falls from my pen it comes to earth i know not when the authors obtain for a brownian particle in a uniform force field the mean and asymptotic first passage times as functions of the particle s initial position and velocity with the recurrence times given as a special case they discuss the region of phase space for which the diffusion model of brownian motion provides an adequate approximation and conclude that there is no possibility of obtaining the recurrence times within that model they find that the nature of the boundary value problem is profoundly altered when the motion is treated as a process in phase rather than configuration space because the time development operator is then parabolic rather than elliptic they argue that such a change in the treatment of brownian motion places it within the sphere of transport theory rather than diffusion theory and that consequently results such as ours have relevance to the study of phenomena such as radiative transfer and neutron transport

half range expansion analysis for langewin dynamics in the high friction limit with a singular absorbing boundary condition noncharacteristic case we consider the dynamics of a brownian particle given by the langevin equation in a strip under the effects of a deterministic force the trajectories of particles originate at a source whose spatial location in the phase space coincides with the location of adsorbing boundaries this leads to singular behavior of trajectories in the high friction limit we use the half range expansion technique and systematic asymptotics to solve a boundary value problem for the fokker planck operator and to calculate the steady state transition probability density the mean time to absorption and the distribution of exit points we do not make assumptions about other parameters in the problem except that they areo 1 relative to the friction coefficient we calculate explicitly the correct location of the milne type extrapolation for absorbing boundary conditions for the smoluchowski approximation to the langevin equation

a linear time algorithm for finding hamiltonian s t paths in even sized rectangular grid graphs with a rectangular hole a grid graph g mathrm g is a finite vertex induced subgraph of the two dimensional integer grid g infty a rectangular grid graph r m n is a grid graph with horizontal size m and vertical size n a rectangular grid graph with a rectangular hole is a rectangular grid graph r m n such that a rectangular grid subgraph r k l is removed from it the hamiltonian path problem for general grid graphs is np complete in this paper we give necessary conditions for the existence of a hamiltonian path between two given vertices in an odd sized rectangular grid graph with a rectangular hole in addition we show that how such paths can be computed in linear time