The reunions of three dissimilar vicious walkers

We study the behavior of three vicious random walkers which diffuse freely in one dimension witharbitrary diffusivitiesb ₁ ² ,b ₂ ² ,b ₃ ² , except that their paths may not cross. The full distribution function is calculated exactly in the continuum limit; the exponent ₃ governing the decay of the probability of a simultaneousreunion of all three walkers aftern steps is found to varycontinuously according to . This variation has consequences for various interfacial wetting transitions in (1+1) dimensions. It may also be related heuristically to the marginality of direct interface-wall interactions decaying asW ₀/l ² in the intermediate fluctuation regime of (1+1)-dimensional wetting, where exponents varying continuously withW ₀ have recently been found.     

Exact distribution of the maximal height of p vicious walkers

Using path-integral techniques, we compute exactly the distribution of the maximal height Hp of p nonintersecting Brownian walkers over a unit time interval in one dimension, both for excursions p watermelons with a wall, and bridges p watermelons without a wall, for all integer p>or=1. For large p, we show that approximately square root 2p (excursions) whereas approximately square root p (bridges). Our exact results prove that previous numerical experiments only measured the preasymptotic behaviors and not the correct asymptotic ones. In addition, our method establishes a physical connection between vicious walkers and random matrix theory.     

Vicious random walkers in the limit of a large number of walkers

The vicious random walker problem on a line is studied in the limit of a large number of walkers. The multidimensional integral representing the probability that thep walkers will survive a timet (denotedP _t ^(p) ) is shown to be analogous to the partition function of a particular one-component Coulomb gas. By assuming the existence of the thermodynamic limit for the Coulomb gas, one can deduce asymptotic formulas forP _t ^(p) in the large-p, large-t limit. A straightforward analysis gives rigorous asymptotic formulas for the probability that after a timet the walkers are in their initial configuration (this event is termed a reunion). Consequently, asymptotic formulas for the conditional probability of a reunion, given that all walkers survive, are derived. Also, an asymptotic formula for the conditional probability density that any walker will arrive at a particular point in timet, given that allp walkers survive, is calculated in the limittp.     

Multiple trapping of random walkers on periodic lattices

Formulas are obtained for the mean absorption time of a set ofk independent random walkers on periodic space lattices containingq traps. We consider both discrete (here we assume simultaneous stepping) and continuous-time random walks, and find that the mean lifetime of the set of walkers can be obtained, via a convolution-type recursion formula, from the generating function for one walker on the perfect lattice. An analytical solution is given for symmetric walks with nearest neighbor transitions onN-site rings containing one trap (orq equally spaced traps), for both discrete and exponential distribution of stepping times. It is shown that, asN , the lifetime of the walkers is of the form Ta_kN², whereT is the average time between steps. Values ofa _k, 2 Sk 6, are provided.     

The principle that generates dissimilar patterns inside aggregates of organisms

Pattern formation and self-organization are phenomena that occur across the board, in animate and inanimate systems. In this paper, we rely on the constructal law to explain the generation of patterns (shapes, structures) in aggregates of organisms-pedestrian crowds and stony corals. In pedestrian crowds a variety of patterns are often observed, from ‘chaotic’ appearances to spontaneous organization in lanes of uniform walking direction. Stony corals and other organisms also present intraspecific variability in shape. We show that flow systems develop in time patterns which provide easier access to the nutrients and space, within a set of constraints imposed by each situation. Flow systems have the freedom to morph their shape in search for architectures that allows them to have greater access to the space that they inhabit. We identify the mechanisms allowing pedestrians to evolve in space and time. We also show that stony corals may develop branched or spherical shapes, depending on which shape performs best in response to the environmental conditions. The constructal law allows systems with complex internal flows to be described and understood for a unified view.     

Coloring of a one-dimensional lattice by two independent random walkers

A new type of question in random walk theory is formulated and solved for the particular case of a periodic one-dimensional lattice. A “red” and a “blue” random walker perform simultaneous independent simple random walk. Each site is initially uncolored and takes irreversibly the color, red or blue, of the first walker by which it is visited. We study the resulting coloring of the final state, in which each site is either red or blue, on a ring of L sites. We calculate the probability P(n, L) that site n is red, in the scaling limit L → ∞ with n/L fixed, for walkers initially on diametrically opposite sites. We determine by simulation the number of interfaces (that is, pairs of neighboring red and blue sites), for initial separation a between the walkers. This number is ≈ 2.5 for initially diametrically opposite walkers, and appears to increase logarithmically with L/a.     

Dynamics of congestion transition triggered by multiple walkers on complex networks

The congestion transition triggered by multiple walkers walking along the shortest path on complex networks is numerically investigated. These networks are composed of nodes that have a finite capacity in analogy to the buffer memory of a computer. It is found that a transition from free-flow phase to congestion phase occurs at a critical walker density f_c, which varies for complex networks with different topological structures. The dynamic pictures of congestion for networks with different topological structures show that congestion on scale-free networks is a percolation process of congestion clusters, while the dynamics of congestion transition on non-scale-free networks is mainly a process of nucleation.     

Vicious random walkers and a discretization of Gaussian random matrix ensembles

The vicious random walker problem on a one-dimensional lattice is considered. Many walkers take simultaneous steps on the lattice and the configurations in which two of them arrive at the same site are prohibited. It is known that the probability distribution of N walkers after M steps can be written in a determinant form. Using an integration technique borrowed from the theory of random matrices, we show that arbitrary kth order correlation functions of the walkers can be expressed as quaternion determinants whose elements are compactly expressed in terms of symmetric Hahn polynomials.     

Capabilities of coupling dissimilar computer codes

A method for automatically controlling commercial computer codes is described. The coupling of the automation system of the EXAFS spectrometer controlled by a DOS-operated computer that we developed and the commercial program of a semiconductor-detector control installed on a Windows-operated computer is given as the example. The described complex system is used for the automation of the intermediate processing of amplitude spectra while measuring EXAFS spectra at the Kurchatov Synchrotron Radiation Source.     

Lattice random walks for sets of random walkers. First passage times

We have studied the mean first passage time for the first of aset of random walkers to reach a given lattice point on infinite lattices ofD dimensions. In contrast to the well-known result ofinfinite mean first passage times for one random walker in all dimensionsD, we findfinite mean first passage times for certain well-specified sets of random walkers in all dimensions, exceptD = 2. The number of walkers required to achieve a finite mean time for the first walker to reach the given lattice point is a function of the lattice dimensionD. ForD > 4, we find that only one random walker is required to yield a finite first passage time, provided that this random walker reaches the given lattice point with unit probability. We have thus found a simple random walk property which sticks atD > 4.Supported in part by a grant from Charles and Renée Taubman and by the National Science Foundation, Grant CHE78-21460.     

Measurement of the Casimir force between dissimilar metals

The first precise measurement of the Casimir force between dissimilar metals is reported. The attractive force, between a Cu layer evaporated on a microelectromechanical torsional oscillator and an Au layer deposited on an Al2O3 sphere, was measured dynamically with a noise level of 6 fN/sqrt[Hz]. Measurements were performed for separations in the 0.2-2 micro m range. The results agree to better than 1% in the 0.2-0.5 micro m range with a theoretical model that takes into account the finite conductivity and roughness of the two metals. The observed discrepancies, which are much larger than the experimental precision, can be attributed to a lack of a complete characterization of the optical properties of the specific samples used in the experiment.     

The three neutrino scenario

I have discussed in my talk several remaining issues in the standard three-flavor mixing scheme of neutrinos, in particular, the sign of Δm²₁₃ and the leptonic CP violating phase. In this report I focus on two topics: (1) supernova method for determining the former sign, and (2) illuminating how one can detect the signatures for both of them in long-baseline ( 10 km) neutrino oscillation experiments. I do this by formulating perturbative frameworks appropriate for the two typical options of such experiments, the high energy and the low energy options with beam energies of 10 GeV and 100 MeV, respectively.     

The first three pomerons...

A model of a three pomeron contribution to high energy elastic pp and scattering is proposed. The data are well described for all momenta ( GeV) and energies ( GeV) (). The model predicts the appearance of two dips in the differential cross-section which will be measured at LHC. The parameters of the pomeron trajectories are . Received: 11 July 2001 / Published online: 20 December 2001     

The three dimensional cavity resonator

The response of a three dimensional cavity to an external excitation is examined. By using the technique of matched asymptotic expansions the full response curve is predicted, and the low frequency Helmholtz mode is studied in some detail. The results are compared with those of Rayleigh and others to reveal some interesting differences.     

Phase diagrams of particles with dissimilar patches: X-junctions and Y-junctions

We use Wertheim's first-order perturbation theory to investigate the phase behaviour and the structure of coexisting fluid phases for a model of patchy particles with dissimilar patches (two patches of type A and f(B) patches of type B). A patch of type α?=?{A,B} can bond to a patch of type β?=?{A,B} in a volume v(αβ), thereby decreasing the internal energy by ?(αβ). We analyse the range of model parameters where AB bonds, or Y-junctions, are energetically disfavoured (?(AB)??0). We show that, for low values of ?(BB)/?(AA), the phase diagram has three different regions: (i)?close to the critical temperature a low-density liquid composed of long chains and rich in Y-junctions coexists with a vapour of chains; (ii)?at intermediate temperatures there is coexistence between a vapour of short chains and a liquid of very long chains with X-?and Y-junctions; (iii)?at low temperatures an ideal gas coexists with a high-density liquid with all possible AA and BB bonds formed. It is also shown that in region (i)?the liquid binodal is reentrant (its density decreases with decreasing temperature) for the lower values of ?(BB)/?(AA). The existence of these three regions is a consequence of the competition between the formation of X-?and Y-junctions: X-junctions are energetically favoured and thus dominate at low temperatures, whereas Y-junctions are entropically favoured and dominate at higher temperatures.     

Using Taguchi method to optimize welding pool of dissimilar laser-welded components

In the present work, CO₂ continuous laser welding process was successfully applied and optimized for joining a dissimilar AISI 316 stainless-steel and AISI 1009 low carbon steel plates. Laser power, welding speed and defocusing distance combinations were carefully selected with the objective of producing welded joint with complete penetration, minimum fusion zone size and acceptable welding profile. Fusion zone area and shape of dissimilar austenitic stainless-steel with ferritic low carbon steel were evaluated as a function of the selected laser welding parameters. Taguchi approach was used as statistical design of experiment (DOE) technique for optimizing the selected welding parameters in terms of minimizing the fusion zone. Mathematical models were developed to describe the influence of the selected parameters on the fusion zone area and shape, to predict its value within the limits of the variables being studied. The result indicates that the developed models can predict the responses satisfactorily.     

Density profile and flow of miscible fluid with dissimilar constituent masses

A computer simulation model is used to study the density profile and flow of a miscible gaseous fluid mixture consisting of differing constituent masses (m_A=m_B/3) through an open matrix. The density profile is found to decay with the height ∝exp (−m_A(B)h), consistent with the barometric height law. The flux density shows a power-law increase ∝(p_c−p)^μ with μ2.3 at the porosity 1−p above the pore percolation threshold 1−p_c.     

Electroelastic behaviour of an annular interfacial crack between dissimilar piezoelectric layers

An annular interfacial crack between dissimilar piezoelectric layers subjected to electroelastic loadings was investigated under an electrically impermeable boundary condition on the crack surface by using the Hankel transform technique and the Cauchy singular integral equation method. The stress intensity factors and energy release rates were determined. Numerical results reveal the effects of crack configuration, electric loads and material parameters on crack propagation and growth. The results should be useful for the design of piezoelectric composite structures and devices of high performance.     

The first three anisotropy constants of nickel

The technique of ferromagnetic resonance at 23 GHz has been used to determine the first three anisotropy constants of pure Ni down to 4.2K. A temperature and orientation dependent linewidth has also been observed.