Aeroacoustics of musical instruments

Musical sound can be generated from numerical solutions obtained from simple physical models of wind-instruments. We call such synthesizers virtual instruments. Crude caricatures capture the global oscillation behaviour of original instruments, providing the musician with a means of expression comparable to that obtained when playing real instruments. Music is, however, produced by details of the flow which correspond to much smaller temporal and spatial scales than the control of the global oscillation. This suggests that one has to split the physical model into two distinct parts: a simple numerical oscillator followed by a complex sound production module, driven by the output of the first module. We illustrate the background of this idea by a discussion of the clarinet, the human voice, and the trombone.General invited lecture presented by A. Hirschberg at the 12th Italian Congress on Theoretical and Applied Mechanics AIMETA '95, Napoli, 3–6 October 1995.     

Another Proof of the Total Positivity of the Discrete Spline Collocation Matrix

We provide a different proof for Morken's result on necessary and sufficient conditions for a minor of the discrete B-spline collocation matrix to be positive and supply intuition for those conditions.     

-complete sequences of integers

An infinite sequence is -complete if every sufficiently large integer is the sum of such that no one divides the other. We investigate -completeness of sets of the form and with nonnegative.

     

Supernumary polylogarithmic ladders and related functional equations

Summary The nature of the polylogarithmic ladder is briefly reviewed, and its close relationship to the associated cyclotomic equation explained. Generic results for the base determined by the family of equationsu ^p +u ^q = 1 are developed, and many new supernumary ladders, existing for particular values ofp andq, are discussed in relation to theirad hoc cyclotomic equations. Results for ordersn from 6 through 9, for which no relevant functional equations are known, are reviewed; and new results for the base , where ³ + = 1, are developed through the sixth order.Special results for the exponentp from 4 through 6 are determined whenever a new cyclotomic equation can be constructed. Only the equationu ⁵+u ³ = 1 has so far resisted this process. The need for the constraint (p,q) = 1 is briefly considered if redundant formulas are to be avoided.The equationu ^6m+1 +u ^6r–1 = 1 is discussed and some valid results deduced. This equation is divisible byu ² –u + 1, and the quotient polynomial is useful for constructing cyclotomic equations. The casem = 1,r = 2 is the first example encountered for which no valid ladders have yet been found.New functional equations to give the supernumary -ladders of index 24 are developed, but their construction runs into difficulty at the third order, apparently requiring the introduction of an adjoint set of variables that blocks the extension to the fourth order.A demonstration, based on the indices of existing accessible and supernumary ladders, indicates that functional equations based on arguments ±z ^m (1–z) ^r (1 +z) ^s are not capable of extension to the sixth order.There are some miscellaneous supernumary ladders that seem incapable, at this time, of analytic proof, and these are briefly discussed. In conclusion, applications of ladders are considered, and attention drawn to the existence of ladders with the base on the unit circle giving rise to Clausenfunction formulas which may play an important role inK-theory.     

Biological chemistry of organotin compounds: Interactions and dealkylation by dithiols

In this paper, we review our past and current efforts toward the elucidation of the biological chemistry of organotin compounds. In particular, we cover two prominent aspects of organotin compounds: their reactivity toward biological dithiols, and their degradation (or metabolization) mechanism using a combination of experimental and computational techniques.     

Molecular mechanics force-field development for amino acid zwitterions

Understanding the conformational flexibility of amino acid zwitterions (ZWs) and their associated conformational energies is crucial for predicting their interactions in biological systems. Gas-phase ab initio calculations of ZWs are intractable. Molecular mechanics (MM), on the other hand, is able to handle large systems but lacks the necessary force field parameters to model ZWs. To develop force field parameters that are able to correctly model ZW geometries and energetics we used a novel combinatorial approach: amino acid ZWs were broken down structurally into key functional components, which were parameterized separately. M?ller-Plesset second-order perturbation calculations on small carboxylates, on the glycine cation, and on novel hydrogen bonded systems, coupled with available experimental data, were used to generate MM3(2000) ZW parameters (Allinger N. L.; Yuh, Y. H.; Lii, J.-H. J Am Chem Soc 1989, 111, 8551). The MM3 results from this combinatorial approach gave geometries that are in good agreement with neutron diffraction experiments, plus their frequencies and energies appear to be reasonably modeled. Current limitations and future development of MM force fields are discussed briefly.     

Stereoselective synthesis of farnesylphosphoryl β-d-arabinofuranose

Decaprenylphosphoryl β-d-arabinofuranose (DPA) is a key arabinose donor in mycobacteria. In an effort to establish a practical synthetic scheme for DPA, the synthesis of nerylphosphoryl and farnesylphosphoryl β-d-arabinofuranoses has been developed. The products were obtained by coupling of a suitably protected β-d-arabinofuranosyl phosphate intermediate with activated forms of the C₁₀ nerol and C₁₅trans,trans-farnesol and subsequent deprotection.     

Searches for fractional electric charge in meteorite samples

The magnetic levitation technique has been used to test for fractional electric charge on samples of several types of meteorite material, thus providing the first mass-independent search for free quarks or other fractionally-charged particles in natural nonterrestrial materials. The tests were made on diced samples of the Hoba and Forsyth County iron-nickel meteorites, and the Murchison stony meteorite, the latter being plated with iron to allow magnetic levitation. No evidence for fractional charge was found, the quantities tested being 1.3 mg (Hoba), 1.1 mg (Forsyth), and 0.4 mg (Murchison).     

Polylogarithmic functional equations: A new category of results developed with the help of computer algebra (MACSYMA)

The interrelation of polylogarithmic functional equations and certain numerical results, known as ladders, is discussed, and leads to a consideration of three new, single-variable functional equations at the second order. Two of these families each contain six leading terms whose interrelationship constitutes a constraint on the integration process, but the third has only a single leading term with no such constraints. It is shown how this functional equation can be integrated to the third order, and the process reduced to an algorithm — actually a sequence of instructions — for incorporation into a computer program for symbolic manipulation. The procedure utilizes results from Kummer's equations to cancel out, in sequence, terms which do not vanish, or do vanish, with the variablez. Arguments are all of the form ±z ^p (1–z) ^q (1+z) ^r , and the process is algebraicized by using a (p,q,r,s) notation (withs=±1) to represent such terms. Application of the procedure leads to an integration to the fourth and fifth orders, the latter exhibiting 55 transcendental terms. The first step for the transition to the sixth order can also be achieved but the subsequent steps are frustrated by the restricted forms that the Kummer equations take at the fifth order — it is not possible to create the needed equations in a form which vanishes withz; this corresponding to the elimination of the (5) constant in the extension of the numerically determined ladders to the sixth and higher orders. The existence of the higher-order ladders strongly suggests functional equations af these orders, but the present process has not yet been successful in finding them. The new equations have, however, produced ladders that were inaccessible from Kummer's equations, and had heretofore been only obtainable numerically, up to the fifth order. The method which was developed should be capable of generalization to other systems of equations characterized by the appearance of arguments with recurrent factors. Some new feature, however, will need to be determined before the barrier to the sixth order can be breached.