In search of Robbins stability

We speculate on whether a certain p-adic stability phenomenon, observed by David Robbins empirically for Dodgson condensation, appears in other nonlinear recurrence relations that "unexpectedly" produce integer or nearly-integer sequences. We exhibit an example (number friezes) where this phenomenon provably occurs.Comment: 10 pages; to appear in the David Robbins memorial issue of Advances in Applied Mathematic

The area of cyclic polygons: Recent progress on Robbins' Conjectures

In his works [R1,R2] David Robbins proposed several interrelated conjectures on the area of the polygons inscribed in a circle as an algebraic function of its sides. Most recently, these conjectures have been established in the course of several independent investigations. In this note we give an informal outline of these developments.Comment: To appear in Advances Applied Math. (special issue in memory of David Robbins

A Robbins-Monro procedure for estimation in semiparametric regression models

This paper is devoted to the parametric estimation of a shift together with the nonparametric estimation of a regression function in a semiparametric regression model. We implement a very efficient and easy to handle Robbins-Monro procedure. On the one hand, we propose a stochastic algorithm similar to that of Robbins-Monro in order to estimate the shift parameter. A preliminary evaluation of the regression function is not necessary to estimate the shift parameter. On the other hand, we make use of a recursive Nadaraya-Watson estimator for the estimation of the regression function. This kernel estimator takes into account the previous estimation of the shift parameter. We establish the almost sure convergence for both Robbins-Monro and Nadaraya--Watson estimators. The asymptotic normality of our estimates is also provided. Finally, we illustrate our semiparametric estimation procedure on simulated and real data.Comment: Published in at http://dx.doi.org/10.1214/12-AOS969 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

The Malkus-Robbins dynamo with a nonlinear motor

In a recent paper Moroz \cite{m02} considered a simplified version of the third class of self-exciting Faraday-disk dynamo model, introduced by Hide \cite{h97}, in the limit in which the Malkus-Robbins dynamo \cite{m72,r77} results as a special case. In that study a linear series motor was incorporated which led to an enriching of the range of possible behaviour that the original Malkus-Robbins dynamo could support. In this paper, we replace the linear motor by a nonlinear motor and consider the consequences on the dynamics of the dynamo

Una nueva interpretación de la definición de Economía de Robbins: los conceptos de escasez real y formal

This article analyses Robbins’s famous definition of economics. It shows that this definition was introduced by the author to solve long-standing problems regarding the subjectmatter of the science that were associated with some of the existing definitions. The article also draws attention to some confusion that surrounds the way Robbins understood the (new) subject-matter and which also slid into his definition. To escape the ambiguities caused by Robbins’s confusion, we propose a more precise way of understanding the subject-matter of economics. The insight gained reveals that Robbins’s definition really contains two sub-definitions: one that describes the subject-matter (real scarcity) and another that describes the method of the science (formal scarcity). This finding sheds light on some analyses and interpretations of this definition in the literatureEste artículo estudia la conocida definición de Economía propuesta por Robbins. En él se muestra, primero, que este autor propone su definición en un intento de resolver algunos problemas inveterados relacionados con la noción de esta ciencia. Seguidamente, el artículo destaca algunas confusiones contenidas en los textos en que Robbins alude a dicha definición. Para resolverlas, se propone aquí un modo más preciso de entender el tema de la ciencia económica. El estudio realizado revela que la definición de Robbins contiene, en verdad, dos sub-definiciones: una que describe el tema de la ciencia (o escasez real) y otra que describe el método de la ciencia (o escasez formal). Este hallazgo permite entender las distintas interpretaciones (a veces contradictorias) de esta definición que existen en la literatur

Quantum indistinguishability from general representations of SU(2n)

A treatment of the spin-statistics relation in nonrelativistic quantum mechanics due to Berry and Robbins [Proc. R. Soc. Lond. A (1997) 453, 1771-1790] is generalised within a group-theoretical framework. The construction of Berry and Robbins is re-formulated in terms of certain locally flat vector bundles over n-particle configuration space. It is shown how families of such bundles can be constructed from irreducible representations of the group SU(2n). The construction of Berry and Robbins, which leads to a definite connection between spin and statistics (the physically correct connection), is shown to correspond to the completely symmetric representations. The spin-statistics connection is typically broken for general SU(2n) representations, which may admit, for a given value of spin, both bose and fermi statistics, as well as parastatistics. The determination of the allowed values of the spin and statistics reduces to the decomposition of certain zero-weight representations of a (generalised) Weyl group of SU(2n). A formula for this decomposition is obtained using the Littlewood-Richardson theorem for the decomposition of representations of U(m+n) into representations of U(m)*U(n).Comment: 32 pages, added example section 4.

Proof of the Refined Alternating Sign Matrix Conjecture

Mills, Robbins, and Rumsey conjectured, and Zeilberger proved, that the number of alternating sign matrices of order $n$ equals $A(n):={{1!4!7! ... (3n-2)!} \over {n!(n+1)! ... (2n-1)!}}$. Mills, Robbins, and Rumsey also made the stronger conjecture that the number of such matrices whose (unique) `1' of the first row is at the $r^{th}$ column, equals $A(n) {{n+r-2} \choose {n-1}}{{2n-1-r} \choose {n-1}}/ {{3n-2} \choose {n-1}}$. Standing on the shoulders of A.G. Izergin, V. E. Korepin, and G. Kuperberg, and using in addition orthogonal polynomials and $q$-calculus, this stronger conjecture is proved.Comment: Plain Te