This paper presents an algorithm for the rapid evaluation of the vector spline in two dimensions. The algorithm is based on the fast multipole method and has an asymptotic CPU time estimate of O(N), where N is the number of data points. The main idea of the algorithm is to combine a cluster of evaluation kernels at diierent data sources into a single kernel in the form of a multipole expansion ...

Long-standing challenges in cluster expansion (CE) construction include choosing how to truncate the expansion and which crystal structures to use for training. Compressive sensing (CS), which is emerging as a powerful tool for model construction in physics, provides a mathematically rigorous framework for addressing these challenges. A recently-developed Bayesian implementation of CS (BCS) pro...

Part 3: (H, PB) Interacting systems, phase transitions and critical phenomena (11 lectures) Interacting spin systems, Ising model. Interacting classical gas, cluster expansion, van der Waals equation of state, Virial Expansion, phase equilibrium, chemical equilibrium. Interacting quantum gases in atom traps. BCS theory of Superconductivity, Landau and Ginzburg Landau theory. Topological excitat...

We extend a T -path expansion formula for arcs on an unpunctured surface to the case of arcs on a once-punctured polygon and use this formula to give a combinatorial proof that cluster monomials form the atomic basis of a cluster algebra of type D.

Benjamini, Lyons and Schramm (1999) considered properties of an infinite graph G, and the simple random walk on it, that are preserved by random perturbations. In this paper we solve several problems raised by those authors. The anchored expansion constant is a variant of the Cheeger constant ; its positivity implies positive lower speed for the simple random walk, as shown by Virág (2000). We ...

Cluster expansions were introduced at the dawn of Statistical Mechanics for the study of high temperature gases of interacting particles. They constitute a powerful perturbative method that is suitable for geometrically large systems, such as encountered in Statistical Physics. Numerous articles have contributed to the subject; a list of relevant publications is [GK, Cam, Bry, KP, Pfi, Dob, BZ,...

Expansion phenomena of laser produced cluster plasma in an external strong magnetic field have been studied. In the cluster plasma expansion process, the ion Larmor radius is often much larger than the initial cluster size, while the electron Larmor radius is smaller than the cluster size, namely, Re < R0 � Ri, where Re(i) is Larmor radius of electron (ion) and R0 is the initial cluster size. W...

CLUSTER EXPANSION WITH APPLICATIONS 3 Assumption 2. There exists a nonnegative function a on X such that for almost all x ∈ X, ∫ d|μ|(y) |ζ(x, y)| e 6 a(x). In order to guess the correct form of a, one should consider the left side of the equation above with a(y) ≡ 0. The integral may depend on x; a typical situation is that x is characterized by a length l(x), which is a positive number, so th...

We introduce a procedure to infer the interactions among a set of binary variables, based on their sampled frequencies and pairwise correlations. The algorithm builds the clusters of variables contributing most to the entropy of the inferred Ising model and rejects the small contributions due to the sampling noise. Our procedure successfully recovers benchmark Ising models even at criticality a...

We have examined the dynamics of noble-gas clusters, heated with high-intensity laser radiation, using pump-probe experiments to temporally resolve the expansion of the clusters. Absorption of the probe radiation is observed to reach a maximum for a particular time delay between pump and probe, dependent on the cluster size. For single-pulse experiments, we find that there is an optimal pulse w...