Is the world made of loops?

I see no good reason to prefer (any version I know of) the `holonomy interpretation' to the `potential interpretation' of the Aharonov-Bohm effect. Everyone agrees that the inverse image $[A]=[A+d\lambda]_{\lambda}=d^{-1}F$ of the electromagnetic field $F$ is a class, full of individuals; and that the circulation $\small{\textsf{C}}$ of the electromagnetic potential $A$ around a loop $\sigma_0$ encircling the solenoid is common to the whole class $[A]$, and to the homotopy class or \emph{hoop} $[\sigma_0]$. If picking individuals out of classes is the problem, picking an individual potential out of $[A]$ should be no worse than picking an individual loop out of $[\sigma_0]$. The individuals of $[A]$ can moreover be transcended---punctually, without integration around loops---by an appropriate version of the electromagnetic connection.Comment: comments welcom

Logic of gauge

The logic of gauge theory is considered by tracing its development from general relativity to Yang-Mills theory, through Weyl's two gauge theories. A handful of elements---which for want of better terms can be called \emph{geometrical justice}, \emph{matter wave}, \emph{second clock effect}, \emph{twice too many energy levels}---are enough to produce Weyl's second theory; and from there, all that's needed to reach the Yang-Mills formalism is a \emph{non-Abelian structure group} (say $\mathbb{SU}\textrm{(}N\textrm{)}$).Comment: comments, corrections most welcom

How Weyl stumbled across electricity while pursuing mathematical justice

It is argued that Weyl's theory of gravitation and electricity came out of `mathematical justice': out of the equal rights direction and length. Such mathematical justice was manifestly at work in the context of discovery, and is enough (together with a couple of simple and natural operations) to derive all of source-free electromagnetism. Weyl's repeated references to coordinates and gauge are taken to express equal treatment of direction and length

Cartesian and Lagrangian momentum

Historical, physical and geometrical relations between two different momenta, characterized here as Cartesian and Lagrangian, are explored. Cartesian momentum is determined by the mass tensor, and gives rise to a kinematical geometry. Lagrangian momentum, which is more general, is given by the fiber derivative, and produces a dynamical geometry. This differs from the kinematical in the presence of a velocity-dependent potential. The relation between trajectories and level surfaces in Hamilton-Jacobi theory can also be Cartesian and kinematical or, more generally, Lagrangian and dynamical

Efficiently Learning from Revealed Preference

In this paper, we consider the revealed preferences problem from a learning perspective. Every day, a price vector and a budget is drawn from an unknown distribution, and a rational agent buys his most preferred bundle according to some unknown utility function, subject to the given prices and budget constraint. We wish not only to find a utility function which rationalizes a finite set of observations, but to produce a hypothesis valuation function which accurately predicts the behavior of the agent in the future. We give efficient algorithms with polynomial sample-complexity for agents with linear valuation functions, as well as for agents with linearly separable, concave valuation functions with bounded second derivative.Comment: Extended abstract appears in WINE 201

Sraffa's Prices

First we consider the existence question in Sraffa’s Chapter I dismissed by counting equations and unknowns. A theorem from the theory of Markov processes, applied to distributions not now of probability but of goods to sectors, shows the general existence of non-negative prices satisfying the conditions imposed by the value equation, that value of output equals value of input. The further condition for these to be unique and positive is that the economy be irreducible, or that no independent sub-economy should exist. Sraffa provides a precise formula determining unique prices, he barely escapes imposing too many conditions on them and certainly cannot require more. In the background and giving motive to the enquiry is the Labour Theory of Value, that goes further. It asserts that the value of anything is ultimately equal to the labour that has gone into making it; so it implies the same principle expressed by the value equation, but a further condition has been added about the nature of the unit. Since the value equation alone makes prices fully determined, there is no room for further conditions, so there is an obstacle to the application of the theory. Standing as a canonical text in a revival of interest in the Theory Of Value serving earlier thought and the later concentration of Ricardo, it offers an exercise in labour value arithmetic, where the only fruit is to find the arithmetic is impossible. An extended interdependence, which applies to repeated production, appears as a stability condition for prices in an adjustment process, and so does the existence of what Sraffa calls a standard commodity, one depending on all others for its production. After treating a case where there is a surplus, and joint production, the relation with Leontief and von Neumann is considered.Schools of Economic Thought and Methodology, Current Heterodox Approaches, Socialist, Marxian, Sraffian

Computation of Consistent Price Indices

The price index, a pervasive long established institution for economics, is a number issued by the Statistical Office that should tell anyone the ratio of costs of maintaining a given standard of living in two periods where prices differ. For a chain of three periods, the product of the ratios for successive pairs must coincide with the ratio for the endpoints. This is the chain consistency required of price indices. A usual supposition is that the index is determined by a formula involving price and quantity data for the two reference periods, as with the one or two hundred in the collection of Irving Fisher, joined with the question of which one to choose and the perplexity that chain consistency is not obtained with any. Hence finally they should all be abandoned. This situation reflects ‘The Index Number Problem’. Now with any number of periods consistent prices indices are all computed together to make a resolution of the ‘Problem’, proved unique and hence never to be joined by others to make a Fisher-like proliferationprice index, price level, index numbers (economics), index number problem

The Price-Level Computation Method

It has been submitted that, for the very large number of different traditional type formulae to determine price indices associated with a pair of periods, which are joined with the longstanding question of which one to choose, they should all be abandoned. For the method proposed instead, price levels associated with periods are first all computed together, subject to a consistency of the data, and then price indices that are as taken together true are determined from their ratios. An approximation method can apply in the case of inconsistency. Here is an account of the mathematics of the methodinflation, index-number problem, non-parametric, price index, price level, revealed preference