Indefinite integrals involving complete elliptic integrals of the third kind

A method developed recently for obtaining indefinite integrals of functions obeying inhomogeneous second-order linear differential equations has been applied to obtain integrals with respect to the modulus of the complete elliptic integral of the third kind. A formula is derived which gives an integral involving the complete integral of the third kind for every known integral for the complete elliptic integral of the second kind. The formula requires only differentiation and can therefore be applied for any such integral, and it is applied here to almost all such integrals given in the literature. Some additional integrals are derived using the recurrence relations for the complete elliptic integrals. This gives a total of 27 integrals for the complete integral of the third kind, including the single integral given in the literature. Some typographical errors in a previous related paper are corrected.     

Indefinite integrals of incomplete elliptic integrals from Jacobi elliptic functions

Integration formulas are derived for the three canonical Legendre elliptic integrals. These formulas are obtained from the differential equations satified by these elliptic integrals when the independent variable u is the argument of Jacobian elliptic function theory. This allows a limitless number of indefinite integrals with respect to the amplitude to be derived for these three elliptic integrals. Sample results are given, including the integrals derived from powers of the 12 Glaisher elliptic functions. New recurrence relations and integrals are also given for the 12 Glaisher elliptic functions.     

Bond stresses for a composite material reinforced with various arrays of fibers

The problem treated here is that of an isotropic body having a doubly periodic rectangular or triangular array of perfectly bonded circular elastic inclusions. The body is in tension or compression. This simulates a composite material wherein a relatively weak matrix is reinforced by stronger (and more rigid) fibers. Bond stresses for both rectangular and triangular arrays have been calculated using either boundary point matching or boundary point least squares techniques. Numerical results based on a plane strain analysis are given in graphical form.     

Genetic algorithms applied to the solution of hybrid optimal control problems in astrodynamics

Many space mission planning problems may be formulated as hybrid optimal control problems, i.e. problems that include both continuous-valued variables and categorical (binary) variables. There may be thousands to millions of possible solutions; a current practice is to pre-prune the categorical state space to limit the number of possible missions to a number that may be evaluated via total enumeration. Of course this risks pruning away the optimal solution. The method developed here avoids the need for pre-pruning by incorporating a new solution approach using nested genetic algorithms; an outer-loop genetic algorithm that optimizes the categorical variable sequence and an inner-loop genetic algorithm that can use either a shape-based approximation or a Lambert problem solver to quickly locate near-optimal solutions and return the cost to the outer-loop genetic algorithm. This solution technique is tested on three asteroid tour missions of increasing complexity and is shown to yield near-optimal, and possibly optimal, missions in many fewer evaluations than total enumeration would require.     

Thou Shalt Not Clone One Bit!

We prove a no-triplets theorem for spin 1 particles, which implies a strengthened form of the no-cloning theorem.     

Indefinite integrals involving the incomplete elliptic integrals of the first and second kinds

A substantial number of indefinite integrals are presented for the incomplete elliptic integrals of the first and second kinds. The number of new results presented is about three times the total number to be found in the current literature. These integrals were obtained with a Lagrangian method based on the differential equations which these functions obey. All results have been checked numerically with Mathematica. Similar results for the incomplete elliptic integral of the third kind will be presented separately.     

The S.M.A.R.T. (small mass,affordable, rapid,transfer-less) digestion method for heavy metal determinations

The S.M.A.R.T. (small mass, affordable, rapid, transfer-less) digestion method was developed to determine heavy metal concentrations in small sample masses. The S.M.A.R.T. digestion method is a hot water bath digestion where sample digestion and dilution are performed in the original sample tube. This method is faster than the typical methods used and reduces potential sources of error. Masses as small as 0.01 g have been digested and analysed using this method. The preparation and digestion time is reduced from 10 h to less than 4 h. Acid volumes are reduced from millilitres to microlitres and the only disposable supplies needed are sample tubes and pipette tips. Method accuracy was determined by digesting seven replicates of two standard reference materials using the S.M.A.R.T. method and analysing samples by inductively coupled plasma mass spectrometry. The S.M.A.R.T. digestion method was found to provide excellent recoveries for Al (76 ± 2.7%), Mn (99 ± 11%), Co (92 ± 17%), Ni (93 ± 28%), Cu (109 ± 33%), Zn (97 ± 7.1%), As (108 ± 20%), Sr (90 ± 12%), Mo (84 ± 23%), Ag (91 ± 1.8%), Cd (95 ± 6.2%), Sn (139 ± 52%) and Pb (95 ± 22%). This study has successfully developed an efficient and reproducible digestion method for heavy metal determination in limited biomass samples.