Radial quadrature for multiexponential integrands

We introduce a Gaussian quadrature, based on the polynomials that are orthogonal with respect to the weight function ln(2)x on the interval [0, 1], which is suitable for the evaluation of radial integrals. The quadrature is exact if the non-Jacobian part of the integrand is a linear combination of a geometric sequence of exponential functions. We find that the new scheme is a useful alternative to existing approaches, particularly for integrands that exhibit multiexponential behavior.     

Liebig's alkaloid analyses: the uncertain route from content to molecular formulae

Liebig's 1831 paper that describes a new apparatus for the analysis of organic compounds and the results of several analyses using the apparatus is a justly famous contribution to the evolution of modern chemistry. In this paper, I look at the three separate components of Liebig's combustion apparatus that collect the water, carbon dioxide and nitrogen released by the combustion of six alkaloids. Gravimetric data included in the paper reveal that very accurate results could be obtained for water by absorption in a calcium chloride tube, and even better results for carbon dioxide resulted from use of the Kaliapparat. Volumetric measurement of nitrogen gave very poor results despite Liebig's efforts to improve it. Inaccuracies in nitrogen measurement made consistent construction of accurate molecular formulae for nitrogenous substances impossible, and only fortuitous decisions intended to bring molecular formulae into agreement with measured combining weights gave formulae in agreement with modern ones, as in the case for quinine.     

A functional equation arising from multiplication of quantum integers

For the quantum integer [n]_q=1+q+q²+?+qⁿ⁻¹ there is a natural polynomial multiplication such that [m]_q⊗_q[n]_q=[mn]_q. This multiplication leads to the functional equation f_m(q)f_n(q^m)=f_mn(q), defined on a given sequence of polynomials. This paper contains various results concerning the construction and classification of polynomial sequences that satisfy the functional equation, as well open problems that arise from the functional equation.     

The Classical Electron Problem

In this paper, we construct a parallel image of the conventional Maxwell theory by replacing the observer-time by the proper-time of the source. This formulation is mathematically, but not physically, equivalent to the conventional form. The change induces a new symmetry group which is distinct from, but closely related to the Lorentz group, and fixes the clock of the source for all observers. The new wave equation contains an additional term (dissipative), which arises instantaneously with acceleration. This shows that the origin of radiation reaction is not the action of a charge on itself but arises from inertial resistance to changes in motion. This dissipative term is equivalent to an effective mass so that classical radiation has both a massless and a massive part. Hence, at the local level the theory is one of particles and fields but there is no self-energy divergence (nor any of the other problems). We also show that, for any closed system of particles, there is a global inertial frame and unique (invariant) global proper-time (for each observer) from which to observe the system. This global clock is intrinsically related to the proper clocks of the individual particles and provides a unique definition of simultaneity for all events associated with the system. We suggest that this clock is the historical clock of Horwitz, Piron, and Fanchi. At this level, the theory is of the action-at-a-distance type and the absorption hypothesis of Wheeler and Feynman follows from global conservation of energy.     

A new efficient presentation for PSL(2,5) and the structure of the groups G(3,m,n)

G(3, m, n) is the group presented by . In this paper, we study the structure of G(3, m, n). We also give a new efficient presentation for the Projective Special Linear group PSL(2, 5) and in particular we prove that PSL(2, 5) is isomorphic to G(3, m, n) under certain conditions.