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Thin film technologies are generally based on plasma-supported methods (see Fig. 4.1). Plasma means

The ability of soluble nonionic surfactants to form liquid expanded (LE) phase is investigated. It is found that the adsorption behavior of nonionic surfactants at liquid interfaces falls into one of two well-defined types: cohesive (with liquid expanded phase) and non-cohesive behavior (without liquid expanded phase). The effect of surfactant structure and properties of the medium on the adsorption parameters of both adsorption types are analyzed. A model of the adsorption constant is proposed, which compares well with experimentally observed values.

This chapter discusses how to claim a major result: how to write it up, and how to release it to the world. Some analogies are drawn from classic Sherlock Holmes stories and films.

In the paper, rapid calculation preprocessing model of particle’s interacting N-body problem is discussed. Coordinate of particle is approximately replayed by grid center. According to the similarity of grid’s topological structure, concept and algorithm of basic solvable range and basic insolvable range of radial distribution function are put forward. In advance, distance function and class partition function between cell centers is found, radial distribution function is computed by translation transformation. In this paper, examples indicate that approximation algorithm is of computing time short and error small, so the approximation algorithm has stronger practicability.

A central problem in computational biology is analyzing genetic sequences to determine the structures of the molecules (RNAs, proteins) they code for. Since this problem is analogous to the problem in computational linguistics of describing what structural descriptions are specified by a given utterance, as first observed by Searls [120], many researchers have tried using formal grammars to analyze biological sequences as well [109, 1, 127, 106]. In this chapter we will attempt a synthesis of this family of approaches, and investigate what properties of formal grammars will determine their success.

We investigate the computational difficulty of approximating the partition function of the ferromagnetic Ising model on a regular matroid. Jerrum and Sinclair have shown that there is a fully polynomial randomised approximation scheme (FPRAS) for the class of graphic matroids. On the other hand, the authors have previously shown, subject to a complexity-theoretic assumption, that there is no FPRAS for the class of binary matroids, which is a proper superset of the class of graphic matroids. In order to map out the region where approximation is feasible, we focus on the class of regular matroids, an important class of matroids which properly includes the class of graphic matroids, and is properly included in the class of binary matroids. Using Seymour's decomposition theorem, we give an FPRAS for the class of regular matroids.

This chapter develops in detail the most advanced of the two coarse graining techniques employed in the previous chapter. Roughly, it consists in looking at the system in blocks of finite size k, which essentially is the annealed correlation length: if we can get suitable estimates for systems up to that size k, we can bound the fractional moment of the partition function of the (arbitrarily large) system in terms of the partition function of a homogeneous model with pinning parameter that depend on the estimates up to size k.

We study the computational complexity of approximately counting the number of independent sets of a graph with maximum degree Δ. More generally, for an input graph G = (V,E) and an activity λ >0, we are interested in the quantity ZG(λ) defined as the sum over independent sets I weighted as w(I) = λ|I|. In statistical physics, ZG(λ) is the partition function for the hard-core model, which is an idealized model of a gas where the particles have non-negibile size. Recently, an interesting phase transition was shown to occur for the complexity of approximating the partition function. Weitz showed an FPAS for the partition function for any graph of maximum degree Δ when Δ is constant and λ 0. We remove the upper bound in the assumptions of Sly's result for Δ ≠ 4, 5, that is, we show that there does not exist efficient randomized approximation algorithms for all λ > λc(TΔ) for Δ = 3 and Δ ≠ 6. Sly's inapproximability result uses a clever reduction, combined with a second-moment analysis of Mossel, Weitz and Wormald which prove torpid mixing of the Glauber dynamics for sampling from the associated Gibbs distribution on almost every regular graph of degree Δ for the same range of λ as in Sly's result. We extend Sly's result by improving upon the technical work of Mossel et al., via a more detailed analysis of independent sets in random regular graphs.

We consider weighted counting of independent sets using a rational weight x: Given a graph with n vertices, count its independent sets such that each set of size k contributes x k . This is equivalent to computation of the partition function of the lattice gas with hard-core self-repulsion and hard-core pair interaction. We show the following conditional lower bounds: If counting the satisfying assignments of a 3-CNF formula in n variables (#3SAT) needs time 2Ω(n) (i.e. there is a c > 0 such that no algorithm can solve #3SAT in time 2 cn ), counting the independent sets of size n/3 of an n-vertex graph needs time 2Ω(n) and weighted counting of independent sets needs time \(2^{\Omega(n/\log^3 n)}\) for all rational weights x ≠ 0.

In this central chapter of Part I we introduce the basic concepts of the RG method invented by Wilson and coauthors in a series of pioneering articles (Wilson 1969, 1971b,c, 1972, Wilson and Fisher 1972, Wilson and Kogut 1974, Wilson 1975).. A very nice introduction to the basic concepts of the Wilsonian RG can also be found in the book by Goldenfeld (1992), which has influenced our presentation.

We begin with a summary of the relevant aspects of molecular line physics. In the next chapter we will present an overview of molecular line astronomy. For more complete accounts of molecular structure, consult the extensive treatments by Herzberg and Herzberg (1960), Bingel (1969), Townes and Schawlow (1975), Gordy and Cook (1970), Flygare (1978), and Kroto (1992).

We consider the one-dimensional Gross-Neveu model and, for a large number N of flavor components, prove the fermion develops a mass dynamically without breakdown of the discrete chiral symmetry.

We discuss a solubility condition for staggered vertex systems with ice-rule and its impacts on the Ashkin-Teller, Potts and layered Potts models.

A non-decreasing sequence of n integers is the degree sequence of a 1-tree (i.e., an ordinary tree) on n vertices if and only if there are least two 1’s in the sequence, and the sum of the elements is 2(n–1). We generalize this result in the following ways. First, a natural generalization of this statement is a necessary condition for k-trees, and we show that it is not sufficient for any k > 1. Second, we identify non-trivial sufficient conditions for the degree sequences of 2-trees. We also show that these sufficient conditions are almost necessary using bounds on the partition function p(n) and probabilistic methods. Third, we generalize the characterization of degrees of 1-trees in an elegant and counter-intuitive way to yield integer sequences that characterize k-trees, for all k.

This Chapter presents recent strategic models of coalition and network formation, with two applications to industrial organization: the formation of cartels and strategic alliances.

Many classes of functional RNA molcules are characterized by highly conserved secondary structures but little detectable sequence similarity. Reliable multiple alignments can therefore be constructed only when the shared structural features are taken into account. Sankoff's algorithm can be used to construct such structure-based alignments of RNA sequences in polynomial time. Here we extend the approach to a probabilistic one by explicitly computing the partition function of all pairwisely aligned sequences with a common set of base pairs. Stochastic backtracking can then be used to compute e.g. the probability that a prescribed sequence-structure pattern is conserved between two RNA sequences. The reliability of the alignment itself can be assessed in terms of the probabilities of each possible match.

We generalise the coalition structure core to partition function games. Our definition relies only on a single crucial assumption, namely that there is some internal consistency in the game. Residuals of the deviation play a game similar to the initial one, and — whenever this is possible — they come to a residual core outcome. Deviating players form their optimistic or pessimistic expectations with this in mind. This leads to a recursive definition of the core. Compared to existing approaches, our core concept displays a reduced sensitivity to behavioural assumptions. We consider the core of an economy with a common pool resource defined by Funaki and Yamato [4], and show that for a number of numerical examples our core concept resolves the puzzle arising when more naive approaches were used. We outline possibilities for further extensions.

Box counting fractal dimensions of Julia sets of the Q-state ferromagnetic Potts model on Bethe lattices are studied. It is shown that Julia sets are circles centered at the origin for values of magnetics field corresponding to the Yang-Lee zeros of the partition function. Also, for values of magnetic field from the Mandelbrot-like set on the complex magnetic field plane Julia sets are simply connected closed Jordan curves and for values of magnetics field outsides the Mandelbrot-like set the Julia set is a Cantor set. Moreover, it is shown that the fractal dimension of a Julia set for a value of magnetic field corresponding to a Yang-Lee zero of the partition function is alocal minimum as a function of magnetic field.

To compute maximum likelihood estimators we must solve the normal equations which amounts to setting the gradient of the likelihood function to zero. For the full likelihood this is a desperate attempt, since expectations with respect to Gibbs measures are involved, and since in general their partition function is not tractable. A popular alternative are pseudolikelihood methods like those originally proposed by J. BESAG and introduced in the last chapter. In Example 18.2.2 we indicated how pseudolikelihood estimators can be computed. On the other hand, expectations can be approximated by Gibbs or Metropolis samplers which gives us a chance to approximate full likelihood estimators.

We have studied quantum effects, structures and phase transitions in Nano-systems. In the following sections an overview is given on the results of our computations on atomic wires, clusters, pore condensates, Bose fluids, elastic properties of model colloids and model colloids in external fields.

We met a variety of distributions on finite products of finite sets of states. They were all strictly positive and only a small number of neighbour sites interacted. These and other common principles will be formalized in the present chapter and illustrated by further examples. It also contains a collection of basic results on random fields.

We discuss a numerical analysis employing the density of partition function zeroes which permits effective distinction between phase transitions of first and second order, elucidates crossover between such phase transitions and gives a new way to measure their strengths in the form of latent heat and critical exponents. Application to a number of models demonstrates the efficacy of the technique.