Lawrence Smolinsky

Citation rates in mathematics: a study of variation by subdiscipline

Variation of citation counts by subdisciplines within a particular discipline is known but rarely systematically studied. This paper compares citation counts for award-winning mathematicians is different subdisciplines of mathematics. Mathematicians were selected for study in groups of rough equivalence with respect to peer evaluation, where this evaluation is given by the awarding of major prizes and grants: Guggenheim fellowships, Sloan fellowships, and National Science Foundation CAREER grants. We find a pattern in which mathematicians working in some subdisciplines have fewer citations than others who won the same award, and this pattern is consistent for all awards. So even after adjustment at the discipline level for different overall citation rates for disciplines, citation counts for different subdisciplines do not match peer evaluation. Demographic and hiring data for mathematics provides a context for a discussion of reasons and interpretations.

A Lie-theoretic Galois theory for the spectral curves of an integrable system. I

In the study of integrable systems of ODEs arising from a Lax pair with a parameter, the constants of the motion occur as spectral curves. The specific curves depend upon the representation of the Lie algebra. In this paper a Galois theory of spectral curves is given that classifies the spectral curves from an integrable system. The spectral curves correspond to conjugacy classes of certain subgroups of the Weyl group for the Lie algebra. The theory is illustrated with the periodic Toda lattice.

Invariants of link cobordism

Abstract This paper is an exposition of some link invariants. These are multisignature invariants associated to nilpotent groups. Their relationship to link cobordism is discussed and an outline of a possible infinite group version is given. These invariants are used to show that there are not link corbodism groups under connected sum.

A generalization of the Levine-Tristram link invariant

Invariants to m-component links are defined and are shown to be link cobordism invariants under certain conditions. Examples are given. In knot and link theory there are certain signature invariants which determine up to torsion a knots cobordism class. These invariants were defined by Tristram [11], Levine [7] and Milnor [81 and are known as the Levine-Tristram or p-signatures. This paper investigates a generalization of the Levine-Tristram signature. The invariant can be viewed as a function which assigns m-roots of unity, one for each link component, to an integer. This invariant is a cobordism invariant in the same sense the Levine-Tristram signature is. However, unlike the Levine-Tristram signature, it is not a weak cobordism invariant. Our approach is geometric along the lines of 0. Ja. Viros geometric interpretation of the Levine-Tristram signature [12]. We discuss the relation of this invariant to the invariant of boundary links defined by Cappell and Shaneson in [2]. Applications of this invariant to the computation of Casson-Gordon invariants has been given in [9 and 10]. The author is grateful to J. Levine for many helpful discussions and to the referee for pointing out two errors. Definitions. An m component link of dimension n or an m-link is an ordered collection of m disjoint smooth oriented submanifolds of S+2 , each of which is homeomorphic to Sn . We assume n > 2 and our links will always be ordered. We will denote a link by (S,+2 ;LI 1 ... , Lm) when we wish to emphasize this point. We also write L = LI U L2 u ... U Lm to denote the link. Every link bounds an oriented manifold called a Seifert surface. If an mlink has an m component Seifert surface so that each component is bounded by exactly one link component then the link is a boundary link. A link is sliced if it bounds a disjoint union of disks in the (n + 3)-ball. A link is weakly sliced if it bounds a disk with punctures. An analysis of the obstructions for a link to be weakly sliced can be carried out for high dimensional links following the usual analysis for knots. Received by the editors July 27, 1987 and, in revised form, March 14, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 57Q45; Secondary 57M25, 57Q60.

Expected number of citations and the crown indicator

The mean normalized citation score or crown indicator is a much studied bibliometric indicator that normalizes citation counts across fields. We examine the theoretical basis of the normalization method and, in particular, the determination of the expected number of citations. We observe a theoretical bias that raises the expected number of citations for low citation fields and lowers the expected number of citations for high citation fields when interdisciplinary publications are included.

A LIE THEORETIC GALOIS THEORY FOR THE SPECTRAL CURVES OF AN INTEGRABLE SYSTEM. II

In the study of integrable systems of ODE’s arising from a Lax pair with a parameter, the constants of the motion occur as spectral curves. Many of these systems are algebraically completely integrable in that they linearize on the Jacobian of a spectral curve. In an earlier paper the authors gave a classification of the spectral curves in terms of the Weyl group and arranged the spectral curves in a hierarchy. This paper examines the Jacobians of the spectral curves, again exploiting the Weyl group action. A hierarchy of Jacobians will give a basis of comparison for flows from various representations. A construction of V. Kanev is generalized and the Jacobians of the spectral curves are analyzed for abelian subvarieties. Prym-Tjurin varieties are studied using the group ring of the Weyl group W and the Hecke algebra of double cosets of a parabolic subgroup of W. For each algebra a subtorus is identified that agrees with Kanev’s Prym-Tjurin variety when his is defined. The example of the periodic Toda lattice is pursued.

Casson-Gordon invariants of some 3-fold branched covers of knots

Abstract In this paper the Casson-Gordon invariants of certain knots are computed using threefold branched covers. These knots provide examples of (4 n +3)-dimensional doubly sliced knots which are not the double of a disk. Examples are also given of (4 n +3)-dimensional knots which are algebraically doubly sliced but not geometrically doubly sliced.

Lax equations, weight lattices, and Prym-Tjurin varieties

The importance of juxtaposing the two approaches to integrable s y s t e m s b y Lie algebras and by algebraic curves -was laid out by Adler and van Moerbeke [AM1], [AM2]. This paper illuminates the interplay of these two ingredients. First, the line bundles on the algebraic curves that give the evolution of the system are shown to be pullbacks of the line bundles of the Borel-Weil theory. Secondly, the Weyl group action on the Jacobian of the master spectral curve (see [MS1], [MS2]) picks out a sub-abelian variety. We show that the flow of the system takes place in this sub-abelian variety. In the periodic Toda lattice, for example, this result applies to F4 and the E-family as well as the bet ter understood A, B, C, D, and G2. This paper is the conclusion of the series [MS1], [MS2]. Here is the setting. Start with a Lax equation, dA/dt= [A, B]. The functions A(s, t) and B(s, t) depend on the t ime t and on a parameter s whose domain is an algebraic curve P. The values of A and B lie in a finite-dimensional Lie algebra and [A, B] is their Lie algebra bracket. Part of the message of [AM1] and [AM2] is that many integrable systems can be writ ten in the form of a Lie algebra-valued Lax equation with a parameter . As is discussed below, we can construct a flow on the Jacobian of the spectral curve which is the normalization of the curve defined by det o(A(s)) z = 0 . This matr ix spectral curve may have several components. Our main theorem considers the flow on the spectral curve associated with the smallest representation as given by the recipe of van Moerbeke and Mumford [MM].

Lotka's inverse square law of scientific productivity: Its methods and statistics

This brief communication analyzes the statistics and methods Lotka used to derive his inverse square law of scientific productivity from the standpoint of modern theory. It finds that he violated the norms of this theory by extremely truncating his data on the right. It also proves that Lotka himself played an important role in establishing the commonly used method of identifying power‐law behavior by the R2 fit to a regression line on a log‐log plot that modern theory considers unreliable by basing the derivation of his law on this very method.

Discrete power law with exponential cutoff and Lotka's law

One of the first bibliometric laws appeared in Alfred J. Lotkas 1926 examination of author productivity in chemistry and physics. The result was a productivity distribution described by a power law. In this paper, Lotkas original data on author productivity in chemistry are reconsidered. We define a discrete power law with exponential cutoff, test Lotkas data, and compare the fit to the discrete power law.