Order Matters

In order to warn undergraduates of a pitfall in linear algebra, let us consider the important case of a matrix eigenvalue case where the basis set is not orthogonal as in 

While we have become accustomed to writing //i = v r, we need to be very careful about the meaning of the indices of the matrices we use and we actually have to think about what we are doing rather than just rearranging matrices. What follows is edited from a set of notes by Prof. James Harrison of the Chemistry Department of Michigan State University. Let us define a basis expansion for i , in terms of a (probably incomplete) set of basis functions Xfi, so that we have 

Here we have used abbreviations io for the orbital index and ib for the basis member index. Let us assume we can write our problem as HC = ESC and focus on just one typical matrix element using inner-index sums for the multiplication steps using row-column multiplication. 

We seek an answer where is a diagonal matrix with all off-diagonal elements equal to zero, which is accomplished by the use of the Kroneker delta = o) = 0(p,/7 a). Then 

That results in (HC), b,io = Sfrb Sx S b, C jo and this does not make any useful sense because the energy is and would refer to some energy of a basis function, not an orbital. Thus we see that using the order HC = ESC does not lead to a useful result. Now consider HC = SCE. 

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How many ways can four students line up in a line, if the order matters ... 

The term permutation is just a fancy way of saying arrangements. When calculating a permutation, you are figuring out the number of possible arrangements. Just remember to use a permutation when precise order matters. 

Choice d is correct. Order matters when writing a ratio. This is because you are comparing one kind of a part to another kind of a part. For example if there was a 20 to 50 ratio of boys to girls, you could write this as 20 50. However, it would be wrong to write 50 20 because there are not 50 boys for every 20 girls. Therefore, a is wrong. If you take the correct ratio 20 50, you are allowed to divide both sides by 10 to get 2 5. Thus choice d is correct. 

Here, the order matters, and you have 8 available places ... 

Aristotle and Plato had reputations of being very wise and knowledgeable men. So most people believed them. Aristotle did not like the randomness of Democritus ideas. He preferred a more ordered matter. Therefore, the idea of atoms and those who believed in their existence had to go underground. 

The second law of thermodynamics applies also to the way we humans order our surroundings. In addition to being complex living systems ourselves, we humans are masters of producing order in the world around us. We manipulate and order matter at the nanoscale level in order to produce the technological breakthroughs that have... 

Used for separating snbscripts in indicating that their order matters... 

Each packet of information follows its own path to the destination. Because of traffic on the network and the number of stops along the way, a packet sent earlier than another packet may arrive later. If the packets are chunks of digitized voice, the order matters, and it is left to the computer at the receiving point to put the packets in the proper order before delivering them to the recipient. 

Formulation consists of mixing the effluent streams coming from the different refining units in order to obtain products conforming to the specifications. It is also at this point that additives are added, the reasons for which and whose action will be described later. One can easily see that as far as octane numbers are concerned, or for that matter any other parameter, the... 

The effect is more than just a matter of pH. As shown in Fig. XV-14, phospholipid monolayers can be expanded at low pH values by the presence of phosphotungstate ions [123], which disrupt the stmctival order in the lipid film [124]. Uranyl ions, by contrast, contract the low-pH expanded phase presumably because of a type of counterion condensation [123]. These effects caution against using these ions as stains in electron microscopy. Clearly the nature of the counterion is very important. It is dramatically so with fatty acids that form an insoluble salt with the ion here quite low concentrations (10 M) of divalent ions lead to the formation of the metal salt unless the pH is quite low. Such films are much more condensed than the fatty-acid monolayers themselves [125-127]. 

Issues associated with order occupy a large area of study for crystalline matter [1, 7, 8]. For nearly perfect crystals, one can have systems with defects such as point defects and extended defects such as dislocations and grain... 

Nomially the amplitude of the total incident field (or intensity of the incident light) is such that the light/matter coupling energies are sufficiently weak not to compete seriously with the dark matter Hamiltonian. As already noted, when this is tire case, tlie induced polarization, P is treated perturbatively in orders of the total electric field. Thus one writes... 

It is only a matter of inserting this hexanomiaT, equation (B 1.3.10). into equation (Bl.3.1) to organize all possible tliree-beam spectroscopies that might appear at any given order. 

Time-resolved fluorescence is perhaps the most direct experunent in the ultrafast spectroscopist s palette. Because only one laser pulse interacts with the sample, the mediod is essentially free of the problems with field-matter time orderings that arise in all of the subsequently discussed multipulse methods. The signal... 

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