Internal Structure of Addition Chains: Well-Ordering

An addition chain for $n$ is defined to be a sequence $(a_0,a_1,\ldots,a_r)$ such that $a_0=1$, $a_r=n$, and, for any $1\le k\le r$, there exist $0\le i, j<k$ such that $a_k = a_i + a_j$; the number $r$ is called the length of the addition chain. The shortest length among addition chains for $n$, called the addition chain length of $n$, is denoted $\ell(n)$. The number $\ell(n)$ is always at least $\log_2 n$; in this paper we consider the difference $\delta^\ell(n):=\ell(n)-\log_2 n$, which we call the addition chain defect. First we use this notion to show that for any $n$, there exists $K$ such that for any $k\ge K$, we have $\ell(2^k n)=\ell(2^K n)+(k-K)$. The main result is that the set of values of $\delta^\ell$ is a well-ordered subset of $[0,\infty)$, with order type $\omega^\omega$. The results obtained here are analogous to the results for integer complexity obtained in [1] and [3]. We also prove similar well-ordering results for restricted forms of addition chain length, such as star chain length and Hansen chain length.Comment: 19 page

Accuracy of simulations for stochastic dynamic models

This paper provides a general framework for the simulation of stochastic dynamic models. Our analysis rests upon a continuity property of invariant distributions and a generalized law of large numbers. We then establish that the simulated moments from numerical approximations converge to their exact values as the approximation errors of the computed solutions converge to zero. These asymptotic results are of further interest in the comparative study of dynamic solutions, model estimation, and derivation of error bounds for the simulated moments

Sperner's problem for G-independent families

Given a graph G, let Q(G) denote the collection of all independent (edge-free) sets of vertices in G. We consider the problem of determining the size of a largest antichain in Q(G). When G is the edge-less graph, this problem is resolved by Sperner's Theorem. In this paper, we focus on the case where G is the path of length n-1, proving the size of a maximal antichain is of the same order as the size of a largest layer of Q(G).Comment: 26 page

ACCURACY OF SIMULATIONS FOR STOCHASTIC DYNAMIC MODELS

This paper provides a general framework for the simulation of stochastic dynamic models. Our analysis rests upon a continuity property of invariant distributions and a generalized law of large numbers. We then establish that the simulated moments from numerical approximations converge to their exact values as the approximation errors of the computed solutions converge to zero. These asymptotic results are of further interest in the comparative study of dynamic solutions, model estimation, and derivation of error bounds for the simulated moments.

Accuracy of simulations for stochastic dynamic models.

This paper provides a general framework for the simulation of stochastic dynamic models. Our analysis rests upon a continuity property of invariant distributions and a generalized law of large numbers. We then establish that the simulated moments from numerical approximations converge to their exact values as the approximation errors of the computed solutions converge to zero. These asymptotic results are of further interest in the comparative study of dynamic solutions, model estimation, and derivation of error bounds for the simulated moments.