The Busy Beaver Competition: a historical survey

Tibor Rado defined the Busy Beaver Competition in 1962. He used Turing machines to give explicit definitions for some functions that are not computable and grow faster than any computable function. He put forward the problem of computing the values of these functions on numbers 1, 2, 3, ... More and more powerful computers have made possible the computation of lower bounds for these values. In 1988, Brady extended the definitions to functions on two variables. We give a historical survey of these works. The successive record holders in the Busy Beaver Competition are displayed, with their discoverers, the date they were found, and, for some of them, an analysis of their behavior.Comment: 70 page

Problems in number theory from busy beaver competition

By introducing the busy beaver competition of Turing machines, in 1962, Rado defined noncomputable functions on positive integers. The study of these functions and variants leads to many mathematical challenges. This article takes up the following one: How can a small Turing machine manage to produce very big numbers? It provides the following answer: mostly by simulating Collatz-like functions, that are generalizations of the famous 3x+1 function. These functions, like the 3x+1 function, lead to new unsolved problems in number theory.Comment: 35 page

Does imperfect competition foster capital accumulation in a developing economy ?

We analyze the relationship between imperfect competition and capital accumulation in a dual economy, with traditional and modern sectors and two types of agents (workers and capitalists). Workers allocate their time endowment between the two sectors. Capitalistsaccumulate wealth in the modern sector. The economy is open to capital flows, but capitalists face borrowing constraints. Non-competitive behavior of capitalists results in a rent which is extracted from the workers and lowers employment in the modern sector. In the longrun,if capitalists are unconstrained, imperfect competition is beneficial for capital accumulation and growth, while it is detrimental in the converse case. Moreover, not-binding borrowing constraints lead to higher employment and wages. This can motivate the introduction of a subsidy on bequests which allows the economy to reach theunconstrained regime, and is welfare-enhancing for workers.imperfect competition; capital accumulation; altruism

Does imperfect competition foster capital accumulation in a developing economy ?

We analyze the relationship between imperfect competition and capital accumulation in a dual economy, with traditional and modern sectors and two types of agents (workers and capitalists). Workers allocate their time endowment between the two sectors. Capitalists accumulate wealth in the modern sector. The economy is open to capital flows, but capitalists face borrowing constraints. Non-competitive behavior of capitalists results in a rent which is extracted from the workers and lowers employment in the modern sector. In the long-run, if capitalists are unconstrained, imperfect competition is beneficial for capital accumulation and growth, while it is detrimental in the converse case. Moreover, not-binding borrowing constraints lead to higher employment and wages. This can motivate the introduction of a subsidy on bequests which allows the economy to reach the unconstrained regime, and is welfare-enhancing for workers.Imperfect competition, capital accumulation, altruism.

Modeling the effects of nuclear fuel reservoir operation in a competitive electricity market

In many countries, the electricity systems are quitting the vertically integrated monopoly organization for an operation framed by competitive markets. In such a competitive regime one can ask what the optimal operation/management of the nuclear generation set is. We place ourselves in a medium-term horizon of the management in order to take into account the seasonal variation of the demand level between winter (high demand) and summer (low demand). A flexible nuclear set is operated to follow a part of the demand variations. In this context, nuclear fuel stock can be analyzed like a reservoir since nuclear plants stop periodically (every 12 or 18 months) to reload their fuel. The operation of the reservoir allows different profiles of nuclear fuel uses during the different seasons of the year. We analyze it within a general deterministic dynamic framework with two types of generation : nuclear and non-nuclear thermal. We study the optimal management of the production in a perfectly competitive market. Then, we build a very simple numerical model (based on data from the French market) with nuclear plants being not operated strictly as base load power plants but within a flexible dispatch frame (like the French nuclear set). Our simulations explain why we must anticipate future demand to manage the current production of the nuclear set (myopia can not be total). Moreover, it is necessary in order to ensure the equilibrium supply-demand, to take into account the non-nuclear thermal capacities in the management of the nuclear set. They also suggest that non-nuclear thermal may remain marginal during most of the year including the months of low demand.Nuclear technology, non-nuclear thermal technology, electricity, nuclear fuel "reservoir", perfect competition, merit order, follow-up of load, seasonal demand.

Rates of convergence for robust geometric inference

Distances to compact sets are widely used in the field of Topological Data Analysis for inferring geometric and topological features from point clouds. In this context, the distance to a probability measure (DTM) has been introduced by Chazal et al. (2011) as a robust alternative to the distance a compact set. In practice, the DTM can be estimated by its empirical counterpart, that is the distance to the empirical measure (DTEM). In this paper we give a tight control of the deviation of the DTEM. Our analysis relies on a local analysis of empirical processes. In particular, we show that the rates of convergence of the DTEM directly depends on the regularity at zero of a particular quantile fonction which contains some local information about the geometry of the support. This quantile function is the relevant quantity to describe precisely how difficult is a geometric inference problem. Several numerical experiments illustrate the convergence of the DTEM and also confirm that our bounds are tight

Modelling the effects of nuclear fuel reservoir operation in a competitive electricity market

In many countries, the electricity systems are quitting the vertically integrated monopoly organization for an operation framed by competitive markets. In such a competitive regime one can ask what the optimal management of the nuclear generation set is. We place ourselves in a medium-term horizon of the management in order to take into account the seasonal variation of the demand level between winter (high demand) and summer (low demand). A flexible nuclear set is operated to follow a part of the demand variations. In this context, nuclear fuel stock can be analyzed like a reservoir since nuclear plants stop periodically (every 12 or 18 months) to reload their fuel. The operation of the reservoir allows different profiles of nuclear fuel uses during the different seasons of the year. We analyze it within a general deterministic dynamic framework with two types of generation: nuclear and non-nuclear thermal. We study the optimal management of the production in a perfectly competitive market. Then, we build a very simple numerical model (based on data from the French market) with nuclear plants being not operated strictly as base load power plants but within a flexible dispatch frame (like the French nuclear set). Our simulations explain why we must anticipate future demand to manage the current production of the nuclear set (myopia can not be total). Moreover, it is necessary in order to ensure the equilibrium supply-demand, to take into account the non-nuclear thermal capacities in the management of the nuclear set. They also suggest that non-nuclear thermal could stay marginal during most of the year including the months of low demand.Electricity Market; nuclear generation; optimal reservoir operation; electricity fuel mix; perfect competition with reservoir

An Empirical Analysis of U.S. Aggregate Portfolio Allocations

This paper analyzes the important time variation in U.S. aggregate portfolio allocations. To do so, we first use flexible descriptions of preferences and investment opportunities to derive optimal decision rules that nest tactical, myopic, and strategic portofolio allocations. We then compare these rules to the data through formal statistical analysis. Our main results reveal that i) purely tactical and myopic investment behaviors are unambiguously rejected, ii) strategic portfolio allocations are strongly supported, and iii) the Fama-French factors best explain empirical portfolios shares.Dynamic Hedging, Risk Aversion, Inter-temporal Substitution, Time-Varying Investment Opportunity Set