Creative Mathematics and the Computer ‐‐ A Correspondence

In this article a pattern arising from the coefficients in the expansion of the sum of the ?th powers of the roots of a quadratic equation is explored by computer. The paper is presented in the form of a correspondence on this problem, which originated during the Computer Appreciation Course held in University College, Galway, in July 1971, for teachers of mathematics for sixth forms. It is felt that the use of the conversational mode may emphasize some of the points of educational interest which arise in the elementary FORTRAN IV programs which are used to generate the array of coefficients. Two methods are adopted: the first involving a direct calculation using binomial coefficients and the second utilizing a recurrence relation. The programs are deliberately unsophisticated so that the educational aspects of the comparison of the methods may be high‐lighted.     

An interative scheme for electronic systems using one-electron green's functions

In this paper an iterative generalization of the minimum principle proposed for electronic systems by Hall, Hyslop, and Rees is investigated. It is shown that this generalization still retains the advantage of using members of a larger class of trial wave-functions, for example those with discontinuities, as initial approximations to the wave-functions. This scheme has the advantage that, at each stage of iteration, an upper bound is obtained which is at least as good as that obtained previously. The theory is first applied to the hydrogen atom. It is then adapted to estimate the Hartree–Fock energy of the helium atom, the Hartree–Fock limit being obtained after a relatively small number of iterations.     

A minimum principle for atomic systems allowing the use of discontinuous wave functions

In this paper a variational principle proposed by Hall [1] is shown to be a minimum principle for coulombic systems. Into this principle it is possible to admit a larger class of trial wave functions than is possible in the conventional variational treatment, including wave functions with discontinuities. It is further shown that the upper bounds given by this treatment are always at least as good as that given by the Rayleigh–Ritz method. The theory is then applied to the hydrogen atom and upper bounds to the energy are calculated for various “cutoff” wave functions. It is usually possible to define an optimum “cut off” distance which minimizes the upper bound.     

Peptide fingerprinting of snake venoms by direct infusion nano-electrospray ionization mass spectrometry: potential use in venom identification and taxonomy

Fingerprinting by mass spectrometry has been increasingly used to study venom variations and for taxonomic analyses based on venom components. Most of these studies have concentrated on components heavier than 3 kDa, but Bothrops snake venoms contain many biologically active peptides, principally C-type natriuretic peptides and bradykinin-potentiating peptides (BPPs). In this work, we have examined the peptide profile of Bothrops venoms (B. alternatus, B. erythromelas, B. insularis, B. jararaca, B. jararacussu, B. leucurus and B. moojeni) using direct infusion nano-electrospray ionization mass spectrometry (nano-ESI-MS) subjecting the data further to principal components analysis (PCA) to assess whether the peptide distributions are reliable in distinguishing the venoms. ESI-MS of a low molar mass fraction obtained by ultrafiltration of each venom (5 kDa nominal cutoff filters) revealed that the venoms have a variety of peptides in common but that each venom also contains taxonomic marker peptides not shared with other venoms. One BPP peptide, QGGWPRPGPEIPP, was found to be common to the seven Bothrops species examined. This peptide may represent a specific marker for this genus since it was not found in the venom of the South American rattlesnake, Crotalus durissus terrificus. PCA on the ESI-MS data reveals a close relationship between B. jararaca, B. jararacussu and B. moojeni venoms, with B. leucurus and B. erythromelas being more distant from these three; B. alternatus and B. insularis were also located distant from these five species, as was C. d. terrificus. These results agree partially with established phylogenetic relationships among these species and suggest that ESI-MS peptide fingerprinting of snake venoms coupled with PCA is a useful tool for identifying venoms and for taxonomic analyses.     

Gadolinium- and superparamagnetic-iron-oxide-enhanced MR findings of intrapancreatic accessory spleen in five patients

Purpose

The purposes of this study were to describe dynamic gadolinium-enhanced magnetic resonance imaging (MRI) findings of intrapancreatic accessory spleen(s) (IPAS) in five patients and to show how superparamagnetic iron oxide (SPIO) enhancement can be used for definite characterization in two cases.

Materials and Methods

An MRI database was searched for patients who had pancreatic tail lesions with imaging features compatible with IPAS between June 2005 and July 2007. Five (four male, one female) patients (age: mean±S.D., 58±9.8 years; range, 50–75 years) were identified. All patients were examined with standard gadolinium-enhanced MRI protocol. Additionally, two patients were examined with SPIO-enhanced MRI protocol. All MRI examinations were retrospectively and blindly evaluated by two radiologists for the predetermined findings, and their final diagnoses were noted.

Results

One pancreatic tail lesion was detected in each patient. All of these lesions were single, focal, well-marginated and located within 3 cm of the distal tail of the pancreas. The mean size (mean±S.D.) of the lesions was (2.02±0.64)×(1.72±0.42) cm², and all lesions had a rounded morphology. The signal intensity of all lesions was similar to that of the spleen on all sequences, including precontrast, postgadolinium and post-SPIO sequences. The reviewers confidently diagnosed IPAS in two patients who had SPIO-enhanced MRI. In the remaining three patients, the reviewers favored the diagnosis of IPAS based on the findings of standard gadolinium-enhanced MRI; however, they could not definitively exclude the other differential diagnoses.

Conclusion

The discovery of a well-marginated, rounded mass in the distal aspect of the tail of the pancreas with signal intensity features of the spleen on all precontrast and postgadolinium sequences suggests the diagnosis of IPAS. However, SPIO-enhanced MRI can be used to characterize the lesion and to establish the definite diagnosis of IPAS in case of clinical doubt.     

The use of slater orbitals in variational calculations involving one-electron green's functions

In this paper an application of a minimum principle proposed for electronic systems by Hall, Hyslop and Rees, [1] and [2], is considered. It is shown that the evaluation of the required two-centre molecular integrals involving one-electron Green's functions may be facilitated by the introduction of a particular class of trial functions. These functions incorporate the potential energy of the system as a weighting factor and Slater-type orbitals are then used as a basis set. The evaluation of the resulting integrals is discussed and illustrative calculations for the H ₂ ⁺ ion are presented.