A regular decomposition of the edge-product space of phylogenetic trees

We investigate the topology and combinatorics of a topological space called the edge-product space that is generated by the set of edge-weighted finite labelled trees. This space arises by multiplying the weights of edges on paths in trees, and is closely connected to tree-indexed Markov processes in molecular evolutionary biology. In particular, by considering combinatorial properties of the Tuffley poset of labelled forests, we show that the edge-product space has a regular cell decomposition with face poset equal to the Tuffley poset.     

偏序集的零调性与可缩性

本文将给出偏序集零调与可缩的几个充分条件,它们包含和深化了文［１］和［２］中的一些结果,把［２］中的一些结果推广到无限偏序集的情形．     

The ‘birth-and-assassination’ process

A new variant of a branching process is introduced, with sufficient conditions for it to persist and to die out. The model is applied to discuss the asymptotic stability of a new type of queuing process.     

Efficient implementation of ‘Optimal’ algorithms in computerized tomography

We describe three optimal algorithms for the reconstruction of a function in R ² from a finite number of line or strip integrals: Optimal recovery, Bayes estimate, Tikhonov-Phillips method. In the case of a rotationally invariant scanning geometry we show that the resulting linear system is a block-cyclic convolution. This observation leads to algorithms which are roughly as efficient as filtered backprojection which is one of the standard methods. The algorithms can be applied to the case of hollow and truncated projections. Numerical examples are given.     

Quantum repulsive Nonlinear Schrödinger models and their ‘Superconductivity’

The fundamental role played by the quantum repulsive Nonlinear Schrödinger (NLS) equation in the evolution of our understanding of the phenomenon of superconductivity in appropriate metals at very low temperatures is surveyed. The first major work was that in 1947 by N. N. Bogoliubov, who studied the very physical 3-space-dimensions problem and super fluidity; and the survey takes the form of an actual dedication to that outstanding scientist who died four years ago. The 3-space-dimensions NLS equation is not integrable either classically or quantum mechanically. But a number of recently discovered closely related lattices in one space dimension (one space plus one time dimension) are integrable as both classical lattices and quantum lattices while their continuum limits are the now well-known fundamental and integrable system the quantum ‘Bose gas’. These models are all examined in this paper in a physical application of recent so-called ‘quantum groups’ theory, itself fundamental to integrability theory. The ‘superfluid’ phase transitions shown by these lattices, as well as by the bose gas, all at zero temperature in 1 + 1 dimensions, are analysed in terms of the behaviour of certain lattice correlation functions which are either quantum or, in the case of the so-called XY-model, classical correlation functions. Although the repulsive NLS models in 1 + 1 are integrable, they do not have actual soliton solutions. Nevertheless the material as surveyed here is a fundamental application of soliton-theory in the broader context of integrability or near-integrability which has had profound effects in the evolution of current understandings in all of modern theoretical physics.     

Propagating wave patterns for the ‘resonant’ Davey–Stewartson system

The resonant nonlinear Schrödinger (RNLS) equation exhibits the usual cubic nonlinearity present in the classical nonlinear Schrödinger (NLS) equation together with an additional nonlinear term involving the modulus of the wave envelope. It arises in the context of the propagation of long magneto-acoustic waves in cold, collisionless plasma and in capillarity theory. Here, a natural (2 + 1) (2 spatial and 1 temporal)-dimensional version of the RNLS equation is introduced, termed the ‘resonant’ Davey–Stewartson system. The multi-linear variable separation approach is used to generate a class of exact solutions, which will describe propagating, doubly periodic wave patterns.     

Global Attractivity and ‘Level-Crossings’ in a Periodic Logistic Integrodifferential Equation

Sufficient conditions are obtained for the existence of a globally attracting positive periodic solution of the logistic integrodifferential equation where a, b are continuous positive periodic functions and K is a nonnegative integrable function defined on {0, ∞). We also derive sufficient conditions for all positive solutions to have “level crossings” about the unique positive periodic solution.     

On the ‘one and one-half dimensional’ relativistic Vlasov–Maxwell system

The time evolution of a collisionless plasma is studied in the case when the Viasov density ? is a function of the time, one space variable and two velocity variables. The electromagnetic fields E, B also have a special structure, and the magnetic field B is non-trivial. It is shown that smooth, consistent initial values generate a uniquc smooth global solution.     

When is ‘nearest neighbour’ meaningful: A converse theorem and implications

Beyer et al. gave a sufficient condition for the high dimensional phenomenon known as the concentration of distances. Their work has pinpointed serious problems due to nearest neighbours not being meaningful in high dimensions. Here we establish the converse of their result, in order to answer the question as to when nearest neighbour is still meaningful in arbitrarily high dimensions. We then show for a class of realistic data distributions having non-i.i.d. dimensions, namely the family of linear latent variable models, that the Euclidean distance will not concentrate as long as the amount of ‘relevant’ dimensions grows no slower than the overall data dimensions. This condition is, of course, often not met in practice. After numerically validating our findings, we examine real data situations in two different areas (text-based document collections and gene expression arrays), which suggest that the presence or absence of distance concentration in high dimensional problems plays a role in making the data hard or easy to work with.