Free index

Every free index represents an increase in the tensor order one free index for vectors, uu two free indices for matrices (second order tensors), r -, three free indices for third order tensors,... 

Keep your eyelids closed but without forcibly blinking. Firmly press your free index finger into the timer comer (near the nose) of the closed eyelids. This will ensure that the eye drop solution remains in the eye and does not drain away immediately into the nose... 

That is, the index i in the r.h.s. term of (A.l) appears twice, which is referred to as the dummy index and implies that the index is added from 1 to 3 in R. On the other hand, we have a free index that appears only once in a term. For example, in the following equation ... 

Topic 5.1.1 Human development index (HDI) and happiness of societies The human development index is used to rank countries by level of human development, and combines normalized measures of life expectancy, literacy, educational attainment, and GDP per capita. To transform a raw variable x into a unit-free index between 0 and 1 (which allows different indices to be added together), the following formula is used ... 

Free index non-repeated subscripts are called free subscripts since they are free to take on any value in 3D space. The count of the free indices on a variable indicates the order of the tensor, e.g. Fj is a vector (first order tensor), Oy is a second order tensor. 

For a free electron gas, it is possible to evaluate the Flartree-Fock exchange energy directly [3, 16]. The Slater detemiinant is constructed using ftee electron orbitals. Each orbital is labelled by a k and a spin index. The Coulomb... 

Free Panicle in ID. The Hamiltonian consists only of the kinetic energy of the particle having mass m ([237] Section 28, [259]). The (unnormalized) energy eigenstates labeled by the momentum index k aie... 

EMBL European Bioinformatics Institute nucleotide sequence database biblio., sub- stance, se- quence 20mio nucleotide seq., 28billion nucleotides journals, author submis- sions European Bioinformatics Institute free daily http //www.e- hi.ac.uk/embl/ index.html... 

The rroo valency index TV is a rneastire of reactivity, especially of free radicals... 

Draw bond order and free valency index diagrams for the butadienyl system. Write a counter into program MOBAS to detemiine how many iterations are executed in solving for the allyl system. The number is not the same for all computers or operating systems. Change the convergence criterion (statement 300) to several different values and determine the number of iterations for each. 

Solvents exert their influence on organic reactions through a complicated mixture of all possible types of noncovalent interactions. Chemists have tried to unravel this entanglement and, ideally, want to assess the relative importance of all interactions separately. In a typical approach, a property of a reaction (e.g. its rate or selectivity) is measured in a laige number of different solvents. All these solvents have unique characteristics, quantified by their physical properties (i.e. refractive index, dielectric constant) or empirical parameters (e.g. ET(30)-value, AN). Linear correlations between a reaction property and one or more of these solvent properties (Linear Free Energy Relationships - LFER) reveal which noncovalent interactions are of major importance. The major drawback of this approach lies in the fact that the solvent parameters are often not independent. Alternatively, theoretical models and computer simulations can provide valuable information. Both methods have been applied successfully in studies of the solvent effects on Diels-Alder reactions. 

In the flask were placed 10.0 g of the propargylic amine (see Chapter lIII-5, Exp. 1). The air in the flask was replaced with nitrogen and a solution of 0.01 mol of KO-tert.-Ci,H,3 in 10 g of THF (free from hydroperoxide) was added. The mixture was warmed at about 40 C. A weakly exothermic reaction was observed and the temperature rose to about 45°C. After 1-2 min the gel originally present, had disappeared almost completely and a brown solution had formed. The refractive index of the solution (note 1) was measured after intervals of about 2 min. After the... 

The products manufactured are predominantiy paraffinic, free from sulfur, nitrogen, and other impurities, and have excellent combustion properties. The very high cetane number and smoke point indicate clean-burning hydrocarbon Hquids having reduced harmful exhaust emissions. SMDS has also been proposed to produce chemical intermediates, paraffinic solvents, and extra high viscosity index (XHVI) lubeoils (see Lubrication and lubricants) (44). 

Chemical modification of the wax can improve smear resistance (5). Sihcones, which do not harm furniture finishes (6), are incorporated as film-forming ingredients in furniture pohshes. The lubricant properties of sihcones improve ease of apphcation of the pohsh and removal of insoluble soil particles. In addition, sihcones make dry films easier to buff and more water-repeUent, and provide depth of gloss, ie, abihty to reflect a coherent image as a result of a high refractive index (7). Wax-free pohshes, which have sihcones as the only film former, can be formulated to dehver smear resistance (8). Another type of film former commonly used in oil-base furniture pohshes is a mineral or vegetable oil, eg, linseed oil. 

The minimum polydispersity index from a free-radical polymerization is 1.5 if termination is by combination, or 2.0 if chains ate terminated by disproportionation and/or transfer. Changes in concentrations and temperature during the reaction can lead to much greater polydispersities, however. These concepts of polymerization reaction engineering have been introduced in more detail elsewhere (6). 

---