Nascent function concepts in Nova Scientia

The origin of the function concept is usually traced to Galileo's work on motion. We argue that specific proto-function concepts appeared in the work of Tartaglia a century before the publication of Galileo's Two New Sciences. The study of Tartaglia's ideas can be used in the classroom as a historical introduction to various function concepts, and certain modern extensions of Tartaglia's optimal range problem and inverse range problem are sources of enrichment for undergraduate courses in analysis, mathematical modeling and computation.     

On rates of convergence for approximations to the generalized inverse

A rate of convergence result for a general class of approximation methods for the generalized inverse of a bounded linear operator with arbitrary range is presented. Applications are given to a number of iterative and noniterative approximation techniques.     

Inversion exercises inspired by mechanics

An elementary calculus transform, inspired by the centroid and gyration radius, is introduced as a prelude to the study of more advanced transforms. Analysis of the transform, including its inversion, makes use of several key concepts from basic calculus and exercises in the application and inversion of the transform provide practice in the use of technology in calculus.     

Inclusions and Identities for the Moore-Penrose Inverse of a Closed Linear Operator

A number of operator inclusions, and identities, involving the Moore-Penrose inverse of a closed densely defined linear operator, are presented. Some of these inclusions generalize known identities in the case when the operator is bounded and has closed range. An application to a known compactness criterion for the Moore-Penrose inverse is also given.     

Computational assessment of the electronic structures of cyclohexa-1,2,4-triene, 1-oxacyclohexa-2,3,5-triene (3delta(2)-pyran), their benzo derivatives, and cyclohexa-1,2-diene. An experimental approach to 3delta(2)-pyran.

The six-membered cyclic allenes given in the title have been studied theoretically by means of an MR-CI approach. For all compounds, the allene structures were found to be the ground states in the gas phase. In the cases of cyclohexa-1,2-diene (1), the isobenzene 2, and the isonaphthalene 7, the most stable structures having a planar allene moiety are the diradicals 1b, 2b, and 7b, representing the transition states for the racemization of 1a, 2a, and 7a and being less stable than the latter by 14.1, 8.9, and 11.2 kcal/mol, respectively. At variance with this order, the 3delta(2)-pyran 4 and the chromene 5 have the zwitterions 4c and 5c as the most stable planar structures, which lie only 1.0 and 5.4 kcal/mol above 4a and 5a, respectively. According to the simulation of the solvent effect, 4c even becomes the ground state of 4 in THF solution. The frontier orbitals of the respective states of 2 and 4 suggest different rates and sites for the reaction with nucleophiles. For the first time, the pyran 4 has been generated and trapped. As a precursor for 4, 3-bromo-4H-pyran (9) was chosen, the synthesis of which was achieved on two routes from 4H-pyran. The treatment of 9 with potassium tert-butoxide (KOt-Bu)/18-crown-6 gave 4-tert-butoxy-4H-pyran as the only discernible product, whether styrene or furan was present, indicating the interception of 4 by KOt-Bu. Finally, the disagreement between the experiment and the theory concerning the heat of formation and the electronic nature of the isobenzene 2 is resolved by demonstrating that the experimental data can provide only an upper limit of the DeltaH(f) degrees value.