We consider a Markov chain of point processes such that each state is a super position of an independent cluster process with the previous state as its centre process together with some independent noise process. The model extends earlier work by Felsenstein and Shimatani describing a reproducing population. We discuss when closed term expressions of the first and second order moments are available for a given state. In a special case it is known that the pair correlation function for these type of point processes converges as the Markov chain progresses, but it has not been shown whether the Markov chain has an equilibrium distribution with this, particular, pair correlation function and how it may be constructed. Assuming the same reproducing system, we construct an equilibrium distribution by a coupling argument.
[(viii)] Then @cite_8 showed that the PCF of @math converges uniformly as @math and he argued that this limiting case may be biologically valid'' [Section 2.4] shimatani:10 . However, there are some flaws in @cite_8 which we address: He did not show that there exists an underlying point process having this limiting case as its PCF, although he claimed that this modified replicated Neyman-Scott process reaches an equilibrium state''. In , for our more general model, we prove the existence of such an underlying point process. When @math is not a stationary Poisson process but its PCF is of a particular form (which we specify later in connection to ), he did not argue that there exists an underlying point process and what it could be. In , we verify this existence under our more general model.
@cite_8
1495091143 2046434141 2001663593 2038883640
Consider a critical branching random walk on the real line. In a recent paper, Aidekon (2011) developed a powerful method to obtain the convergence in law of its minimum after a log-factor translation. By an adaptation of this method, we show that the point process formed by the branching random walk seen from the minimum converges in law to a decorated Poisson point process. This result, confirming a conjecture of Brunet and Derrida (J Stat Phys 143:420–446, 2011), can be viewed as a discrete analog of the corresponding results for the branching Brownian motion, previously established by (2010, 2011) and (2011). Consider a graph, G, for which the vertices can have two modes, 0 or 1. Suppose that a particle moves around on G according to a discrete time Markov chain with the following rules. With (strictly positive) probabilities pm, pc and pr it moves to a randomly chosen neighbour, changes the mode of the vertex it is at or just stands still, respectively. We call such a random process a (pm, pc, pr)-lamplighter process on G. Assume that the process starts with the particle in a fixed position and with all vertices having mode 0. The convergence rate to stationarity in terms of the total variation norm is studied for the special cases with G = KN, the complete graph with N vertices, and when G = mod N. In the former case we prove that as N --> [infinity], ((2pc + pm) 4pcpm)N log N is a threshold for the convergence rate. In the latter case we show that the convergence rate is asymptotically determined by the cover time CN in that the total variation norm after aN2 steps is given by P(CN > aN2). The limit of this probability can in turn be calculated by considering a Brownian motion with two absorbing barriers. In particular, this means that there is no threshold for this case. par>We prove some non-approximability results for restrictions of basic combinatorial optimization problems to instances of bounded “degreeror bounded “width.” Specifically: We prove that the Max 3SAT problem on instances where each variable occurs in at most B clauses, is hard to approximate to within a factor @math , unless @math . H stad [18] proved that the problem is approximable to within a factor @math in polynomial time, and that is hard to approximate to within a factor @math . Our result uses a new randomized reduction from general instances of Max 3SAT to bounded-occurrences instances. The randomized reduction applies to other Max SNP problems as well. We observe that the Set Cover problem on instances where each set has size at most B is hard to approximate to within a factor @math unless @math . The result follows from an appropriate setting of parameters in Feige's reduction [11]. This is essentially tight in light of the existence of @math -approximate algorithms [20, 23, 9] We present a new PCP construction, based on applying parallel repetition to the inner verifier,'' and we provide a tight analysis for it. Using the new construction, and some modifications to known reductions from PCP to Hitting Set, we prove that Hitting Set with sets of size B is hard to approximate to within a factor @math . The problem can be approximated to within a factor B [19], and it is the Vertex Cover problem for B =2. The relationship between hardness of approximation and set size seems to have not been explored before. We observe that the Independent Set problem on graphs having degree at most B is hard to approximate to within a factor @math , unless P = NP . This follows from a comination of results by Clementi and Trevisan [28] and Reingold, Vadhan and Wigderson [27]. It had been observed that the problem is hard to approximate to within a factor @math unless P = NP [1]. An algorithm achieving factor @math is also known [21, 2, 30, 16 . We prove that, in the hierarchy of simple types based on the type of natural numbers, any finite strongly stable function is equal to the application of the semantics of a PCF-definable functional to some strongly stable (generally not PCF-definable) functionals of type two. Applying a logical relation technique, we derive from this result that the strongly stable model of PCF is the extensional collapse of its sequential algorithms model.