The on-axis magnetic well and Mercier's criterion for arbitrary stellarator geometries

A simplified analytical form of the on-axis magnetic well and Mercier's criterion for interchange instabilities for arbitrary three-dimensional magnetic field geometries is derived. For this purpose, a near-axis expansion based on a direct coordinate approach is used by expressing the toroidal magnetic flux in terms of powers of the radial distance to the magnetic axis. For the first time, the magnetic well and Mercier's criterion are then written as a one-dimensional integral with respect to the axis arclength. When compared with the original work of Mercier, the derivation here is presented using modern notation and in a more streamlined manner that highlights essential steps. Finally, these expressions are verified numerically using several quasisymmetric and non-quasisymmetric stellarator configurations including Wendelstein 7-X.

Analytical Interpretation of the Early Literature on the Theory of Vibrations

A discussion is given of early literature pertaining to the theory of vibration from the time of Pythagoras up through 1750. The paper attempts to give an analytical interpretation to early anecdotal works concerning Pythagoras and to publications of Galileo, Huygens, Hooke, Taylor, John Bernoulli, Leibniz and Euler. To bridge the “culture gap,” mathematical developments by the latter cited authors are, whenever appropriate, rephrased in modern notation, using, for the most part, only those techniques that should have been well known to the authors at the time. The emphasis is on what might be loosely called the physics (or the mathematical physics) of vibration.

Notation and Transcription

Using as its basis one source of Heinrich Isaac’s “Güretzsch,” Chapter 16 gives a step-by-step guide on how to transcribe into modern notation a piece of music in renaissance notation. This chapter presents “translations” of all the elements of original notation, including note values, rests, clefs, and time signatures. We also offer practical suggestions on how to rehearse when using partbooks in manuscript or original print.

Performance Practice of Early American Hymnody: Tempo and the “Moods of Time”

Modern time signatures indicate metrical organization in notated music. However, in most American hymnals and psalters published between 1721 and 1809, time signatures also signify very specific tempi. This notational practice, further removed from modern usage than any other element of this music, derives from proportional notations abandoned in art music in the seventeenth century. As technically complex music was published using this notation in the 1760s, these time signatures began to be used more subtly. In combination, they provide metrical effects unlike those possible with modern time signatures: doubling or halving tempo, or maintaining the pulse while altering its division or larger metric organization. Viewed from the perspective of modern notation, these functions diverge from their appearance. This article clarifies the correlation between time signature and tempo indicated in eighteenth- and early nineteenth-century American tunebooks (hymnals), arguing for its inclusion in modern performances of this repertoire. Internal evidence and related pedagogical practices suggest these tempi were intended to be observed; most early American theorists, composers, and compilers advocated adherence. Any revival of repertoire first published in this notation, including the works of such composers as William Billings, Daniel Read, and Supply Belcher, would profit by observing these tempi. In a repertoire frequently devoid of interpretive markings, time signatures provide invaluable clues to performers.

Kleene's Amazing Second Recursion Theorem

This little gem is stated unbilled and proved (completely) in the last two lines of §2 of the short note Kleene [1938]. In modern notation, with all the hypotheses stated explicitly and in a strong (uniform) form, it reads as follows:Second Recursion Theorem (SRT). Fix a set V ⊆ ℕ, and suppose that for each natural number n ϵ ℕ = {0, 1, 2, …}, φn: ℕ1+n ⇀ V is a recursive partial function of (1 + n) arguments with values in V so that the standard assumptions (a) and (b) hold with.(a) Every n-ary recursive partial function with values in V is for some e.(b) For all m, n, there is a recursive function : Nm+1 → ℕ such that.Then, for every recursive, partial function f of (1+m+n) arguments with values in V, there is a total recursive function of m arguments such thatProof. Fix e ϵ ℕ such that and let .We will abuse notation and write ž; rather than ž() when m = 0, so that (1) takes the simpler formin this case (and the proof sets ž = S(e, e)).

Types in Logic and Mathematics Before 1940

In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead's Principia Mathematica ([71], 1910–1912) and Church's simply typed λ-calculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege's Grundgesetze der Arithmetik for which Russell applied his famous paradox and this led him to introduce the first theory of types, the Ramified Type Theory (RTT). We present RTT formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from RTT leading to the simple theory of types STT. We present STT and Church's own simply typed λ-calculus (λ→C) and we finish by comparing RTT, STT and λ→C.

On the Gauss mean-value formula for class number

In his masterwork Disquisitiones Arithmeticae, Gauss stated an approximate formula for the average of the class number for negative discriminants. In this paper we improve the known estimates for the error term in Gauss approximate formula. Namely, our result can be written as for every ∊ > 0, where H(−n) is, in modern notation, h(−4n). We also consider the average of h(−n) itself obtaining the same type of result.Proving this formula we transform firstly the problem in a lattice point problem (as probably Gauss did) and we use a functional equation due to Shintani and Dirichlet class number formula to express the error term as a sum of character and exponential sums that can be estimated with techniques introduced in a previous work on the sphere problem.