Rapid mixing of Gibbs sampling on graphs that are sparse on average

Gibbs sampling also known as Glauber dynamics is a popular technique for sampling high dimensional distributions defined on graphs. Of special interest is the behavior of Gibbs sampling on the Erd?s‐Rényi random graph G(n,d/n), where each edge is chosen independently with probability d/n and d is fixed. While the average degree in G(n,d/n) is d(1 ‐ o(1)), it contains many nodes of degree of order log n/log log n. The existence of nodes of almost logarithmic degrees implies that for many natural distributions defined on G(n,p) such as uniform coloring (with a constant number of colors) or the Ising model at any fixed inverse temperature β, the mixing time of Gibbs sampling is at least n^{1+Ω(1/log log n)}. Recall that the Ising model with inverse temperature β defined on a graph G = (V,E) is the distribution over {±}^Vgiven by . High degree nodes pose a technical challenge in proving polynomial time mixing of the dynamics for many models including the Ising model and coloring. Almost all known sufficient conditions in terms of β or number of colors needed for rapid mixing of Gibbs samplers are stated in terms of the maximum degree of the underlying graph. In this work, we show that for every d < ∞ and the Ising model defined on G (n, d/n), there exists a β_d > 0, such that for all β < β_d with probability going to 1 as n →∞, the mixing time of the dynamics on G (n, d/n) is polynomial in n. Our results are the first polynomial time mixing results proven for a natural model on G (n, d/n) for d > 1 where the parameters of the model do not depend on n. They also provide a rare example where one can prove a polynomial time mixing of Gibbs sampler in a situation where the actual mixing time is slower than npolylog(n). Our proof exploits in novel ways the local tree like structure of Erd?s‐Rényi random graphs, comparison and block dynamics arguments and a recent result of Weitz. Our results extend to much more general families of graphs which are sparse in some average sense and to much more general interactions. In particular, they apply to any graph for which every vertex v of the graph has a neighborhood N(v) of radius O(log n) in which the induced sub‐graph is a tree union at most O(log n) edges and where for each simple path in N(v) the sum of the vertex degrees along the path is O(log n). Moreover, our result apply also in the case of arbitrary external fields and provide the first FPRAS for sampling the Ising distribution in this case. We finally present a non Markov Chain algorithm for sampling the distribution which is effective for a wider range of parameters. In particular, for G(n, d/n) it applies for all external fields and β < β_d, where d tanh(β_d) = 1 is the critical point for decay of correlation for the Ising model on G(n, d/n). © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Complete assignments of the 1H and 13C NMR spectra of 9,10-didehydro retinoic acids

A new class of retinoic acids was synthesized containing a 9,10-rigid bond. 1H and 13C NMR spectra were assigned for eight new compounds (all-E, 13Z) containing a carboxylic acid or tertiobutylester polar end group. Assignments were based on the combination of one- and two-dimensional experiments (APT, COSY, HMBC).     

Annulated 1,2,3,4-tetrahydro-β-carbolines by intramolecular Mannich-type amino- and amidoalkylations of N(9)-(ω-nitroalkyl)-3,4-dihydro-β-carbolines

2-[1-(ω-Nitroalkyl)-1H-indol-3-yl]ethylformamides 11 were transformed to the corresponding 9-(ω-nitroalkyl)-4,9-dihydro-3H-β-carbolines 5 and through a diastereoselective intramolecular aminoalkylation to the annulated tetrahydro-β-carbolines 13, in high yields. Intramolecular N-acyliminium cyclisation of compounds 5 afforded the tetracyclic diazacycloalkano[jk]fluorenes in two diastereoisomeric forms 18 and 19 with moderate selectivity. Conjugate addition reactions performed on compounds 18 and 19 led to pentacyclic indolo[3,2,1-de]pyrido[3,2,1-jk]naphthyridinone 26a or diazabenzo[a]naphtho[2,1,8-cde]azulenone 26b.     

Algebraic properties of some new vector-valued rational interpolants

In a recent paper of the author [A. Sidi, A new approach to vector-valued rational interpolation, J. Approx. Theory, 130 (2004) 177–187], three new interpolation procedures for vector-valued functions F(z), where , were proposed, and some of their properties were studied. In this work, after modifying their definition slightly, we continue the study of these interpolation procedures. We show that the interpolants produced via these procedures are unique in some sense and that they are symmetric functions of the points of interpolation. We also show that, under the conditions that guarantee uniqueness, they also reproduce F(z) in case F(z) is a rational function.     

The cardinals below

The results of this paper concern the effective cardinal structure of the subsets of [ω₁]^<ω₁, the set of all countable subsets of ω₁. The main results include dichotomy theorems and theorems which show that the effective cardinal structure is complicated.     

Spanning trails containing given edges

A graph G is Eulerian-connected if for any u and v in V(G), G has a spanning (u,v)-trail. A graph G is edge-Eulerian-connected if for any e^′ and e^″ in E(G), G has a spanning (e^′,e^″)-trail. For an integer r?0, a graph is called r-Eulerian-connected if for any X⊆E(G) with |X|?r, and for any , G has a spanning (u,v)-trail T such that X⊆E(T). The r-edge-Eulerian-connectivity of a graph can be defined similarly. Let θ(r) be the minimum value of k such that every k-edge-connected graph is r-Eulerian-connected. Catlin proved that θ(0)=4. We shall show that θ(r)=4 for 0?r?2, and θ(r)=r+1 for r?3. Results on r-edge-Eulerian connectivity are also discussed.     

Resolvable group divisible designs with block size four and index three

In this paper, we investigate the existence of resolvable group divisible designs (RGDDs) with block size four, group-type hⁿ and index three. The necessary conditions for the existence of such a design are n?4 and hn≡0. These necessary conditions are shown to be sufficient except for (h,n)∈{(2,4),(2,6)} and possibly excepting (h,n)=(2,54).     

Pair lengths of product graphs

The pair length of a graph G is the maximum positive integer k, such that the vertex set of G can be partitioned into disjoint pairs {x,x^′}, such that d(x,x^′)?k for every x∈V(G) and x^′y^′ is an edge of G whenever xy is an edge. Chen asked whether the pair length of the cartesian product of two graphs is equal to the sum of their pair lengths. Our aim in this short note is to prove this result.     

Solutions of nonlinear evolution inclusions

In this paper, the solution of the nonlinear evolution inclusion problem of the form u^′(t)+B(t,u(t))∋f(t) is studied. In this problem, the operators are of type (M) or type (S₊), which are different from those of pseudo-monotone operators that had been studied by many authors. At the same time, we study the perturbation problem. In fact, many kinds of evolution equations can be generalized by this problem. The former results are improved and generalized by our conclusions, and we will give more applications.