Tempered expansion

In certain plant habitats or niches, access to resources depends crucially upon rapid growth under conditions of climatic stress. Examples of this phenomenon are particularly obvious on shallow soils in continental climates where the growth window between winter cold and summer desiccation may be extremely short. In deciduous woodlands in the cool temperate zone an essentially similar niche arises in the period between snow melt and closure of the tree canopy. Both circumstances provide opportunities for high rates of photosynthesis and mineral nutrient capture in the late spring but depend upon rapid expansion of roots and shoots in the low-temperature conditions of the late winter and early spring. 

The rate of convergence of expansions in the basis (1.2) has received little attention except for purely numerical studies [3,7,8,9,16] which indicated that the convergence is at least (unlike for bais set of type) not frustratingly slow. Rather detailed studies were performed for the even-tempered basis set, i.e. for exponents constructed from two parameters and /di (for each /)... 

The key feature is - both for the expansion of 1/r or e " in terms of even-tempered Gaussians - that, for large n, the cut-off error goes as exp(-anh) with h the step size and that the discretization errors goes as exp(—6/h), with a and b constants. While - for fixed n a small h is good for the discretization error, it is bad for the cut-off error and vice versa. The best compromise is that h 1/%/ni which implies that the overall error goes as exp(—Cy )-... 

We were able to show analytically - in an unexpectedly tricky way (the mathematical ingredients of which are in the appendix) - that the error of an expansion of the function 1/r in terms of an even-tempered Gaussian basis of dimension n goes as exp(-ci/n) provided that the two parameters of the even-tempered basis are optimized. 

For quantum chemistry the expansion of e in a Gaussian basis is, of course, much more important than that of 1/r. The formalism is a little more lengthy than for 1/r, but the essential steps of the derivation are the same. For an even-tempered basis one has a cut-off error exp(—n/i) and a discretization error exp(-7//i), such that results of the type (2.15) and (2.16) result. Of course, e is not well represented for r very small and r very large. This is even more so for 1/r, but this wrong behaviour has practically no effect on the rate of convergence of a matrix representation of the Hamiltonian. This is very different for basis set of type (1.1). Details will be published elsewhere. 

Of all the marine ecosystems, temperate estuaries seem to be undergoing the most degradation. Evidence for this is seen in (1) increased sedimentation and turbidity, (2) more extended and expansive episodes of hypoxia or anoxia, (3) loss of seagrasses and dominant suspension feeders, primarily bivalves, and (4) a higher frequency and duration of nuisance and toxic algal blooms, jellyfish infestations, and fish kills. Overall, these... 

Cooler heads must have come into play to temper this expansiveness because there is no evidence that the proposal was ever presented to or approved by the ACSI or the Secretary of the Army. It was clear that, on many counts, the project had violated Department of Defense (DOD) and Department of the Army (DA)... 

Table 8 Second-order many-body perturbation theory corrections to beryllium-like ions using non-relativistic (E ), Dirac-Coulomb (E ) and Dirac-Coulomb-Breit (E ) hamiltonians, obtained using the atomic precursor to BERTHA, known as SWIRLES. Basis sets are even-tempered S-spinors of dimension N= 17, with exponent sets, Xi generated by Xi = abi-i, with a = 0.413, and p = 1.376. Angular momenta in the range 0 < / < 6 have been included in the partial wave expansion of each second-order energy, and the total relativistic correction toE has been collected as Ef. All energies in hartree.

The expansion of marine fisheries into tropical waters has increased the risks of widespread human intoxications in temperate latitudes. This situation has been facilitated by increased travel, rapid transport of food stuffs, and improved refrigeration facilities. 

Due both to the expansion of the psychopharmacol-ogical armamentarium and the advances in the medical treatment of severely ill pediatric patients, child and adolescent psychiatrists are increasingly being called upon to manage these patients adjunctively with psychiatric medication. This need has been tempered with concerns over the potential effects of psychotropic medications on the developing central nervous system (CNS) (Jensen, 1998 Vitiello, 1998). The management of these medically ill children and adolescents falls into several categories ... 

Reciprocating expanders, in normal operation, should not accept liquid in any form during the expansion cycle. However, the reciprocating device can tolerate some liquid for short periods of time provided that none of the constituents freeze out in the expander cylinder and cause serious mechanical problems. If selected design conditions indicate possibilities of entering the liquid and especially the triple point range on expansion during normal operation, then inlet pressure and temper-... 

A third technique to produce tempered ware uses two glasses (or ceramics) formed together (laminated), each with different thermal coefficient of expansion. This is used to make Correll dishes by Coming. For this tableware, a pyrocer-amic material with one type of thermal coefficient of expansion is covered with another pyroceramic with a greater thermal coefficient of expansion and is then baked until the outer layer melts uniformly. Materials with greater thermal coefficients of expansion will expand more when heated and will contract more when cooled. The greater contraction (once cooled) of the outside material causes compression. 

Figure 22.15 presents coefficients of expansion for a range of materials of different temper, all taken at a cooling rate of 0.1 C/min. As would be expected from the cooling curve data, there is little difference between samples with different temper level, whilst those of untempered material are significantly different from tempered samples. The peak for the tempered material is at about 23 C, whilst that of untempered material is at about 14 C. The data for untempered material does however show a small peak in the coefficient of expansion at 23 C, indicating that some (albeit not very much) material has been tempered. This was unexpected, and illustrates the sensitivity of the method. 

Figure 22.14. Coefficient of expansion vs temperature for well tempered chocolate cooled at different rates (Pinschower 2003).

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