We study self-avoiding and neighbour-avoiding walks and lattice trails on two semiregular lattices, the (3.122) lattice and the (4.82) lattice. For the (3.122) lattice we find the exact connective constant for both self-avoiding walks, neighbour-avoiding walks and trails. For the (4.82) lattice we generate long series which permit the accurate estimation of the connective constant for self-avoi...

In this short note we revisit the gain of integrability property of the gain part of the Boltzmann collision operator. This property implies the W l,r k regularity propagation for solutions of the associated space homogeneous initial value problem. We present a new method to prove the gain of integrability that simplifies the technicalities of previous approaches by avoiding the argument of gai...

Many, if not most, commercially available diabetes treatment products are protected by some form of intellectual property. This article discusses the development and commercialization of products in view of the state of intellectual property for the diabetes treatment market, with respect to possible discouragement, for some, from seeking patent protection or commercializing a new product under...

Concave compositions are compositions (i.e. ordered partitions) of a number in which the parts decrease up to the middle summand(s) and increase thereafter. Perhaps the most surprising result is for even length, concave compositions where the generating function turns out to be the quotient of two instances of the pentagonal number theorem with variations of sign. The false theta function disco...

We prove that the self-avoiding walk has Ornstein-Zernike decay and some related properties for all noncritical temperatures at which the model is defined.

This paper proves the formula

We derive the Brownian bridge asymptotics for a scaled self-avoiding walk conditioned on arriving to a far away point n~a for ~a ∈ (Z, 0, ..., 0), and outline the proof for all other ~a in Zd.

Probabilistic properties are considered as the most important requirements for a variety of software systems, since they are used to formulate extra-functional requirements such as reliability, availability, safety, security and performance requirements. Currently, several probabilistic logics have been proposed to specify such important properties. However, due to the inherent complexity of th...

A self-avoiding walk (saw) is a path on a lattice that does not pass through the same point twice. Though mathematicians have studied saws for over fifty years, the number of n-step saws is unknown. This paper examines a special case of this problem, finding the number of nstep “up-side” saws (ussaws), saws restricted to moving up and sideways. It presents formulas for the number of n-step ussa...