By H. Crapo (auth.), H. Crapo, D. Senato (eds.)

This booklet, devoted to the reminiscence of Gian-Carlo Rota, is the results of a collaborative attempt via his associates, scholars and admirers. Rota was once one of many nice thinkers of our occasions, innovator in either arithmetic and phenomenology. i believe moved, but touched by way of a feeling of unhappiness, in featuring this quantity of labor, regardless of the phobia that i'll be unworthy of the duty that befalls me. Rota, either the scientist and the guy, used to be marked by way of a generosity that knew no bounds. His principles opened vast the horizons of fields of analysis, allowing an marvelous variety of scholars from everywhere in the globe to turn into enthusiastically concerned. The contagious power with which he tested his large psychological means consistently proved clean and encouraging. past his renown as talented scientist, what was once quite extraordinary in Gian-Carlo Rota was once his skill to understand the various highbrow capacities of these earlier than him and to evolve his communications therefore. This human experience, complemented by means of his acute appreciation of the significance of the person, acted as a catalyst in bringing forth some of the best in every one of his scholars. Whosoever was once lucky sufficient to get pleasure from Gian-Carlo Rota's longstanding friendship used to be such a lot enriched by means of the event, either mathematically and philosophically, and had celebration to understand son cote de bon vivant. The booklet opens with a heartfelt piece by way of Henry Crapo within which he meticulously items jointly what Gian-Carlo Rota's premature loss of life has bequeathed to science.

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**Example text**

47 Step 5. Extend by linearity. This completes the definition of the linear functional E. We next COlle to the most disquieting feature of umbral notation. Let f(a, {3, x) and g(a, {3, x) be two polynomials in the variables a, {3, x. We write f(a, {3, x) ~ g(a, {3, x) to mean E(f(a, Read ~ f3, x» = E(g(a, f3, x». as "equivalent to". The "classics" went a bit too far, they wrote f(a, f3, x) = g(a, {3, x) that is, they replaced the symbol ~ by ordinary equality. This was an excessive abuse of notation.

We will answer both these questions simultaneously. Let us go back to threedimensional space. You all know that the set of all straight lines in space - not necessarily through the origin - forms a nice algebraic variety, called the Grassmannian. The group of all Euclidean rigid motions acts on the Grassmannian, and there is an invariant measure on the Grassmannian under the action of the group of Euclidean motions. This invariant measure is unique except for a constant factor. A similar statement may be made about the set of all planes, and more generally for the set of all linear varieties of dimension k in Euclidean space of dimension n.

Einstein wrote: "Common sense is the residue of those prejudices that were instilled into us before the age of seventeen". Common sense must constantly readjust to reality. The new measure IL I that we obtain in this way is called the mean width, a misnomer that has been kept for historical reasons. The mean width of a solid in space is completely characterized by Axioms I, 2, 3, and 4/1. In particular, it is invariant, that is, it does not depend on position. For example, the formula for the mean width of a sphere of radius r is computed to be Thus we see that in three dimensions each of the three elementary symmetric functions of three variables leads to an invariant measure that enjoys equal rights with volume.