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The algebra of operators is noncommutative, so that all of the ordinary algebra, calculus, and analysis with ordinary numbers becomes of small utility for operators. Thus, for a single operator, a, ordinary functions of this operator, such as A = exp a, can be defined, for example, by power series. These functions obey the rules of ordinary analysis even though a is an operator. But if another operator 13 is introduced with which a does not commute, the question of functions of the two variables a, (3 is beset with commutation difficulties and the simplest theorems of analysis are lost. For example, if B = exp 13, it is not true that BA, that is exp 13 exp a, is equal to exp(f3 + a). -Richard Phillips Feynman [200]

This is a very useful tool to disentan$le exponentials of certain operato~s ~ften appear in quantum optics.) Suppose that A and B are two operators such that [A, BJ =

If Items for which you have created a purchase order arrive ahead of the bill and/or invoice, you can either create an Item receipt and pay for them when the invoice comes or pay for them immediately using the instructions in the previous section.

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Electronic whiteboard A virtual whiteboard that enable multiple collabo rators to take turns authoring and modifying hand-drawn or computer-generated graphics, highlighting points of interest on digital images, or presenting a digitized slide as part of a presentation Entrez The search and retrieval system that integrates information from the National Center for Biotechnology (NCBI) databases Enzymes Proteins that make chemical reactions take place faster Eugenics Selective breeding or mating; the improvement of the human species by encouraging or permitting reproduction of only those indi viduals with genetic characteristics judged desirable Functional genomics The study of genes, their resulting proteins, and the role played by the proteins in the body s biochemical processes G8 The eight most industrialized nations (the Russian Federation, United States, Canada, United Kingdom, France, Germany, Italy, and Japan) GenBank An annotated collection of all publicly available DNA se quences provided by the National Institutes of Health (NIH).

more general disentanglement techniques, look at [142,200, 369]

6, where 6 commutes with both A and B. Then the Baker-Hausdorffformuia holds:

To record and account for the Items when you receive them, follow these steps: 1. Go to Vendors Receive Items. The Create Item Receipts window shown in Figure 17-5 appears. 2. Choose the vendor from whom you received the Items. 3. Click Yes in the Open PO s Exist window. 4. In the Open Purchase Orders window, select each PO that contains Items you re receiving, and then click OK.

a (Aa + B(3) = exp (B(3) exp (Aa) exp ((32

Gene The fundamental physical and functional unit of heredity A gene is an ordered sequence of nucleotides located in a particular position on a particular chromosome that de nes a speci c RNA molecule Gene chip A device that supports rapid, large-scale genetic analysis based on a large array of gene probes on a specially treated glass slide or other substrate Thousands of experiments can be run on an array the size of a dime Also referred to as microarray Gene mapping Determination of the relative positions of genes on a DNA molecule and of the distance, in linkage units or physical units, between them Gene testing Analyzing an individual s genetic material to determine pre disposition to a particular condition or to con rm a diagnosis of ge netic disease.

6) .

5. In the Create Item Receipts window, make any necessary changes to the line items, such as changing the quantity or cost (or deleting an Item completely from the detail area). 6. Save the Item receipt.

(D.l)

Notice that, in general, a function of Aa + B(3 can always be written as a sum of products of a function of Aa, a function of B(3, and a function of 6. What the BakerHausdorff formula states that is not immediately obvious is that when the function of Aa + B(3 is an exponential, all the other functions are exponentials, too, and moreover, there is just one term in the sum. To see that it is really so, let us take (3, A,

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