Social Security Administration : actions taken to strengthen procedures for issuing Social Security numbers to noncitizens but some weaknesses remain : report to the Chairman, Subcommittee on Social Security, Committee on Ways and Means, House of Representatives /

"GAO-04-12"

Detailed annual report of the Registrar General for Ireland containing a general abstract of the numbers of marriages, births, and deaths registered in Ireland.

Title varies slightly.

Hearing to review the U.S. Department of Agriculture's release of program beneficiaries' social security numbers and the department's information systems, generally : hearing before the Committee on Agriculture, House of Representatives, One Hundred Tenth Congress, first session, May 2, 2007.

Shipping list no.: 2009-0305-P.

Crowgey's Universal calculator, and general international accountant; containing a table of algarisms, or series of numbers, in duplex arithmetical progression ...

Mode of access: Internet.

The reader : containing I. The art of delivery ... a selection of lessons in the various kinds of prose : II. Poetick numbers ... a selection of lessons in the various kinds of verse : being the third part of a Columbian exercise, the whole comprising an easy and systematical method of teaching and of learning the English language /

Mode of access: Internet.

List of merchant vessels of the United States, with the official numbers and signal letters awarded to them.

Report year ends June 30.

The making of index numbers; a study of their varieties, tests, and reliability,

"Selected bibliography": p. 519-520.

The old farmer and his almanack; being some observations on life and manners in New England a hundred years ago, suggested by reading the earlier numbers of Mr. Robert B. Thomas's Farmer's almanack, together with curious, instructive, and entertaining, as well as a variety of miscellaneous matter,

Mode of access: Internet.

World prices and the building industry; index numbers of prices of 40 basic commodities for 14 countries in currency and in gold, and material on the building industry,

Mode of access: Internet.

1960 price list and supplement to volume 1 of Criswell's currency series : current market value on notes listed by Criswell's numbers.

Mode of access: Internet.

Complex numbers and symmetries in quantum mechanics, and a nonlinear superposition principle for Wigner functions

Complex numbers appear in the Hilbert space formulation of quantum mechanics, but not in the formulation in phase space. Quantum symmetries are described by complex, unitary or antiunitary operators defining ray representations in Hilbert space, whereas in phase space they are described by real, true representations. Equivalence of the formulations requires that the former representations can be obtained from the latter and vice versa. Examples are given. Equivalence of the two formulations also requires that complex superpositions of state vectors can be described in the phase space formulation, and it is shown that this leads to a nonlinear superposition principle for orthogonal, pure-state Wigner functions. It is concluded that the use of complex numbers in quantum mechanics can be regarded as a computational device to simplify calculations, as in all other applications of mathematics to physical phenomena.

Tacit collusion, firm asymmetries and numbers:evidence from EC merger cases

This paper estimates the implicit model, especially the roles of size asymmetries and firm numbers, used by the European Commission to identify mergers with coordinated effects. This subset of cases offers an opportunity to shed empirical light on the conditions where a Competition Authority believes tacit collusion is most likely to arise. We find that, for the Commission, tacit collusion is a rare phenomenon, largely confined to markets of two, more or less symmetric, players. This is consistent with recent experimental literature, but contrasts with the facts on ‘hard-core’ collusion in which firm numbers and asymmetries are often much larger.

Tacit vs. overt collusion firm asymmetries and numbers:what's the evidence?

It is conventional wisdom that collusion is more likely the fewer firms there are in a market and the more symmetric they are. This is often theoretically justified in terms of a repeated non-cooperative game. Although that model fits more easily with tacit than overt collusion, the impression sometimes given is that ‘one model fits all’. Moreover, the empirical literature offers few stylized facts on the most simple of questions—how few are few and how symmetric is symmetric? This paper attempts to fill this gap while also exploring the interface of tacit and overt collusion, albeit in an indirect way. First, it identifies the empirical model of tacit collusion that the European Commission appears to have employed in coordinated effects merger cases—apparently only fairly symmetric duopolies fit the bill. Second, it shows that, intriguingly, the same story emerges from the quite different experimental literature on tacit collusion. This offers a stark contrast with the findings for a sample of prosecuted cartels; on average, these involve six members (often more) and size asymmetries among members are often considerable. The indirect nature of this ‘evidence’ cautions against definitive conclusions; nevertheless, the contrast offers little comfort for those who believe that the same model does, more or less, fit all.