Social interaction as a heuristic for combinatorial optimization problems

We investigate the performance of a variant of Axelrod's model for dissemination of culture - the Adaptive Culture Heuristic (ACH) - on solving an NP-Complete optimization problem, namely, the classification of binary input patterns of size $F$ by a Boolean Binary Perceptron. In this heuristic, $N$ agents, characterized by binary strings of length $F$ which represent possible solutions to the optimization problem, are fixed at the sites of a square lattice and interact with their nearest neighbors only. The interactions are such that the agents' strings (or cultures) become more similar to the low-cost strings of their neighbors resulting in the dissemination of these strings across the lattice. Eventually the dynamics freezes into a homogeneous absorbing configuration in which all agents exhibit identical solutions to the optimization problem. We find through extensive simulations that the probability of finding the optimal solution is a function of the reduced variable $F/N^{1/4}$ so that the number of agents must increase with the fourth power of the problem size, $N \propto F^ 4$, to guarantee a fixed probability of success. In this case, we find that the relaxation time to reach an absorbing configuration scales with $F^ 6$ which can be interpreted as the overall computational cost of the ACH to find an optimal set of weights for a Boolean Binary Perceptron, given a fixed probability of success

On the size of the fibers of spectral maps induced by semialgebraic embeddings

Let ${\mathcal S}(M)$ be the ring of (continuous) semialgebraic functions on a semialgebraic set $M\subset{\mathbb R}^m$ and ${\mathcal S}^*(M)$ its subring of bounded semialgebraic functions. In this work we compute the size of the fibers of the spectral maps ${\rm Spec}({\tt j})_1:{\rm Spec}({\mathcal S}(N))\to{\rm Spec}({\mathcal S}(M))$ and ${\rm Spec}({\tt j})_2:{\rm Spec}({\mathcal S}^*(N))\to{\rm Spec}({\mathcal S}^*(M))$ induced by the inclusion ${\tt j}:N\hookrightarrow M$ of a semialgebraic subset $N$ of $M$. The ring ${\mathcal S}(M)$ can be understood as the localization of ${\mathcal S}^*(M)$ at the multiplicative subset ${\mathcal W}_M$ of those bounded semialgebraic functions on $M$ with empty zero set. This provides a natural inclusion ${\mathfrak i}_M:{\rm Spec}({\mathcal S}(M))\hookrightarrow{\rm Spec}({\mathcal S}^*(M))$ that reduces both problems above to an analysis of the fibers of the spectral map ${\rm Spec}({\tt j})_2:{\rm Spec}({\mathcal S}^*(N))\to{\rm Spec}({\mathcal S}^*(M))$. If we denote $Z:={\rm cl}_{{\rm Spec}({\mathcal S}^*(M))}(M\setminus N)$, it holds that the restriction map ${\rm Spec}({\tt j})_2|:{\rm Spec}({\mathcal S}^*(N))\setminus{\rm Spec}({\tt j})_2^{-1}(Z)\to{\rm Spec}({\mathcal S}^*(M))\setminus Z$ is a homeomorphism. Our problem concentrates on the computation of the size of the fibers of ${\rm Spec}({\tt j})_2$ at the points of $Z$. The size of the fibers of prime ideals `close' to the complement $Y:=M\setminus N$ provides valuable information concerning how $N$ is immersed inside $M$. If $N$ is dense in $M$, the map ${\rm Spec}({\tt j})_2$ is surjective and the generic fiber of a prime ideal ${\mathfrak p}\in Z$ contains infinitely many elements. However, finite fibers may also appear and we provide a criterium to decide when the fiber ${\rm Spec}({\tt j})_2^{-1}({\mathfrak p})$ is a finite set for ${\mathfrak p}\in Z$.Comment: 33 pages, 3 figure

Strong statistical stability of non-uniformly expanding maps

We consider families of transformations in multidimensional Riemannian manifolds with non-uniformly expanding behavior. We give sufficient conditions for the continuous variation (in the $L^1$-norm) of the densities of absolutely continuous (with respect to the Lebesgue measure) invariant probability measures for those transformations.Comment: 21 page

To Begin the World Over Again: A Reimagining of Millennial Expectations in Colonial America as Source to the Revolution

The study’s controlling question is to determine the extent millennialism as an intellectual movement informed the thinking of Colonial America. The evidence gathered suggests a kind of smorgasbord with no uniform thought. What can be deduced from the literature, however, is the fluidity of millennialism and its ability to adapt and contort to the political ideology of the era. Moreover, millennialism provided a sense of purpose for the American continent with the Great Awakening serving as the legitimizing movement which both popularized and diffused the millennium. From the 1750s, American millennialism began its evolution from a spiritual consummation of all things to a more politicized millennium, mirroring England. By the time of the Revolution, millennial themes, symbols, and language dominated the American continent far more than Whig ideology