Jesper Lützen

Heaviside s Operational Calculus and the Attempts to Rigorise It

At the end of the 19th century Oliver Heaviside developed a formal calculus of differential operators in order to solve various physical problems. The pure mathematicians of his time would not deal with this unrigorous theory, but in the 20th century several attempts were made to rigorise Heavisides operational calculus. These attempts can be grouped in two classes. The one leading to an explanation of the operational calculus in terms of integral transformations (Bromwich, Carson, Vander Pol, Doetsch) and the other leading to an abstract algebraic formulation (Lévy, Mikusiński). Also Schwartzs creation of the theory of distributions was very much inspired by problems in the operational calculus.

Interactions between mechanics and differential geometry in the 19th century

Conclusion79. This study of the interaction between mechanics and differential geometry does not pretend to be exhaustive. In particular, there is probably more to be said about the mathematical side of the history from Darboux to Ricci and Levi Civita and beyond. Statistical mechanics may also be of interest and there is definitely more to be said about Hertz (I plan to continue in this direction) and about Poincarés geometric and topological reasonings for example about the three body problem [Poincaré 1890] (cf. also [Poincaré 1993], [Andersson 1994] and [Barrow-Green 1994]). Moreover, it would be interesting to find out how the 19th century ideas discussed here influenced the developments in the 20th century. Einstein himself is a hotly debated case.Yet, despite these shortcommings, I hope that this paper has shown that the interactions between mechanics and differential geometry is not a 20th century invention. Kleins view (see my Introduction) that Riemannian geometry grew out of mechanics, more specifically the principle of least action, cannot be maintained. On the other hand, when Riemannian geometry became known around 1870 it was immediately used in mechanics by Lipschitz. He began a continued tradition in this field, which had several elements in common with the new view of mechanics conceived by the physicists and explicitly carried out by Hertz.Before 1870 we found only scattered interactions between differential geometry and mechanics and only direct ones for systems of two or three degrees of freedom. For more degrees of freedom the geometrical ideas were in some interesting cases taken over by analogy, but these analogies did not lead to formal introduction of geometries of more than three dimensions.

Julius Petersen 1839–1910 a biography

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. The person Julius Petersen Childhood and youth (1839-1871) Geometric constructions (1866-1879) The doctoral dissertation (1871) Social and economic engagement (1871-1877) Cryptography (1875) The theory of algebraic equations (1877) Docent at the Polytechnical School. Mechanics (1871-1887) Style of exposition and method of research Miscellaneous papers (1870-1890) Models and instruments (1887-1895) Invariant theory and graph theory (1888-1899) Professor at Copenhagen University (1887-1909) Inspector of the Learned Schools (1887-1900) Function theory, latin squares and number theory (1888-1909) Last years (1908-1910) References 9 12 14 18 23 28 29 34 37 38 47 49 67 69 75 78 79

Joseph Liouville's work on the figures of equilibrium of a rotating mass of fluid

After Jacobis surprising discovery in 1834 that rotating triaxial ellipsoids of fluid could be in equilibrium, Joseph Liouville (1809–1882) began a study of the properties of these figures and of the well known equilibrium ellipsoids of revolution found by Maclaurin. He published six papers on the question, but only a small fraction of his most far-reaching investigations on the stability of the figures of equilibrium, made during the last months of 1842, appeared in print.This paper contains an almost complete reconstruction of Liouvilles theory of stability in its historical context. It is based on two manuscripts published here for the first time and on numerous calculations in Liouvilles notebooks. Liouvilles idea is to determine whether the “force vive” (the kinetic energy) of a perturbed equilibrium figure has a maximum in the equilibrium state. That he did by expanding the perturbations in a series of Lamé functions.Being largely unpublished, Liouvilles theory of stability of equilibrium figures had little impact. Neverthelss the published notes were known to one of his successors, Liapounoff, and the oher successor, Poincaré, was indirectly influenced by Liouvilles work on Lamé functions. Liouville had published this work without revealing that it had been developed as a tool for his investigations of the stability of rotating ellipsoids.

Joseph Liouville's contribution to the theory of integral equations

Abstract It is well known that Liouville did pioneering work on the application of specific integral equations in different parts of mathematics and mathematical physics. However, his short paper on spectral theory of Hilbert-Schmidt like operators has been neglected. With that paper Liouville initiated the general theory of integral equations.

The Physical Origin of Physically Useful Mathematics

Abstract When Wigner claimed that the effectiveness of mathematics in the natural sciences was unreasonable it was due to a dogmatic formalist view of mathematics according to which higher mathematics is developed solely with a view to formal beauty. I shall argue that this philosophy is not in agreement with the actual practice of mathematics. Indeed, I shall briefly illustrate how physics has influenced the development of mathematics from antiquity up to the twentieth century. If this influence is taken into account, the effectiveness of mathematics is far more reasonable.

Julius Petersen annotated bibliography

Abstract This is the first comprehensive bibliography of Julius Petersens papers and books, covering not only his mathematical works, but also his contributions to economics, social science, physics and education.

The Mathematical Correspondence between Julius Petersen and Ludvig Sylow

The correspondence between the two Scandinavian mathematicians Julius Petersen and Ludvig Sylow is interesting, not only because it sheds light on the researches of the two correspondents, but also because it gives a good impression of the general state of affairs in algebra, more specifically group theory and Galois theory during the early and middle 1870’s, as seen from the nothern fringe of the European mathematical scene. It thus complements Hans Wussing’s vivid and detailed account of this period in his classical The Genesis of the Abstract Group Concept [Wussing 1969/84]. Wussing shows that around 1870 the mathematical avantgarde had emancipated group theory as a separate discipline independent of its use in the theory of equations, thereby paving the way for the introduction of the abstract group concept around 1880. The correspondence between Petersen and Sylow, however, reminds us of the fact that it takes time for such conceptual innovations to filter down through the system. Indeed, for Petersen and Sylow, as well as many of their colleagues, groups were still tied to equations, if they were known at all, and the abstract definition was a far cry of the future. Waterhouse [1980], and in particular Schar-Lau [1988], have emphasized that Sylow’s proof of the famous theorems named after him, built crucially on the fact that he considered groups as Galois groups of a certain equation. In this connection, Scharlau quotes the laconic proof in Petersen’s algebra book [Petersen 1877] of the theorem stating that every group can be considered as a Galois group.

The interaction of physics, mechanics and mathematics in Joseph Liouville's research

As many of his contemporaries did, Joseph Liouville often emphasized the importance of physics for mathematical research. His own works provide a host of examples of interactions between mathematics and physics. This paper analyses some of them. It is shown how Laplacian physics gave rise to Liouville’s theory of differentiation of arbitrary order, how Kelvin’s research on electrostatics gave rise to Liouville’s theorem about conformal mappings and how the theory of heat conduction gave rise to Sturm-Liouville theory. It will be shown how the problem of the shape of the planets was an important inspiration for Liouville’s far reaching studies of Lame functions and spectral theory of a particular type of integral operators. Finally the interactions between Liouville’s work on mechanics and differential geometry will be discussed.

Let G be a group

Traditional philosophy of mathematics has dealt extensively with the question of the nature of mathematical objects such as number, point, and line. Considerations of this question have a great interest from a historical point of view, but they have become largely outdated in light of the development of modern mathematics. The prevalent view of mathematics in the 20th (and the early part of the 21st) century has been some variant of the so-called formalistic view, according to which mathematics is a study of axiomatic systems or structures. In such structures the objects have exactly the properties set out in the axioms and nothing else can be said about their nature. The question of the nature of mathematical objects has become a non-issue. Similarly, the structure itself is completely determined by its set of axioms and one can say nothing meaningful about its nature other than that. In particular, in the formalistic understanding of mathematics a structure exists if its axioms are consistent among each other (i.e., non-contradicting).