Symmetric Functions in Noncommuting Variables

Consider the algebra Q> of formal power series in countably many noncommuting variables over the rationals. The subalgebra Pi(x_1,x_2,...) of symmetric functions in noncommuting variables consists of all elements invariant under permutation of the variables and of bounded degree. We develop a theory of such functions analogous to the ordinary theory of symmetric functions. In particular, we define analogs of the monomial, power sum, elementary, complete homogeneous, and Schur symmetric functions as will as investigating their properties.Comment: 16 pages, Latex, see related papers at http://www.math.msu.edu/~sagan, to appear in Transactions of the American Mathematical Societ

Does Duality Theory Hold in Practice? A Monte Carlo Analysis for U.S. Agriculture

The Neoclassical theory of production establishes a dual relationship between the profit value function of a competitive firm and its underlying production technology. This relationship, usually referred to as the duality theory, has been widely used in empirical work to estimate production parameters without the requirement of explicitly specifying the technology. We analyze the ability of this approach to recover the underlying production parameters and its effects on estimated elasticities and scale economies measurements, when data available for estimation features typical realistic problems. We design alternative scenarios and compute the data generating process by Monte Carlo simulations, so as to know the true technology parameters as well as to calibrate the dataset to yield realistic magnitudes of noise. This noise introduced in the estimation by construction prevents duality theory from holding exactly. Hence, the true production parameters may not be recovered with enough precision, and the estimated elasticities or scale economies measurements may be more inaccurate than expected. We compare the estimated production parameters with the true (and known) parameters by means of the identities between the Hessians of the production and profit functions.duality theory, firm’s heterogeneity, measurement error, data aggregation, omitted variables, endogeneity, uncertainty, Monte Carlo simulations., Crop Production/Industries, Production Economics, Risk and Uncertainty, Q12, D22, D81,

Towards a Democratization of Knowledge with Topological Emphasis in Economics

Abstract. We formulate and prove a theorem which consists in how the natural endogenous antagonist interaction of agents who look for understanding a generalizable phenomenon, results in a tendency towards chaos. This takes us to the final absolution of implementing the majority rule as the only instrument that generates socially acceptable knowledge, escaping from the chaos tendency. Finally, we extend our analysis to consider the arise of multiple simultaneous antagonist postures on the explanation of a phenomenon, and through an application of the Pythagoras theorem, we prove that it takes less effort or sacrifice for an agent to learn strategically to get an explanation, than if she was the creator of the concerning knowledge, which implies different consequences of possible topological private and public tendencies.Keywords. Antagonist endogenous knowledge, Social entropy, Chaos theorem, Social choice.JEL. B50, O31, O35

Emptiness existence: A free-strategic view

Abstract. We formulate a theorem to always keep within mind on the emptiness existence followed by its matematicall proof, that takes as a base a single axiom named hipoteticity i.e. an element exists iff it has a structure.Keywords. Emptiness, Existence, Coalitions, Game Theory.JEL. C70, C71 ,C72, C79

Necessary conditions for the positivity of Littlewood-Richardson and plethystic coefficients

We give necessary conditions for the positivity of Littlewood-Richardson coefficients and SXP coefficients. We deduce necessary conditions for the positivity of the plethystic coefficients. Explicitly, our main result states that if $S^\lambda(V)$ appears as a summand in the decomposition into irreducibles of $S^\mu(S^\nu(V))$, then $\nu$'s diagram is contained in $\lambda$'s diagram.Comment: 11 pages, 7 figure