Bracket notation

The M-th quantum mechanical state can be described independently of its representation by employing Dirac s nofflh on, Ym) = n). This nota- 

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Here we have adopted a Dirac bracket notation which should be distmguished from the ensemble... 

Hereafter, we shall regularly use the square brackets notation, [ ], to indicate concentration. Thus, we read [Fe+I] as ferric ion concentration. ... 

Now let us rewrite the wave Equation 7.1 in what is known as a Dirac s bracket notation ... 

Some synonyms in the Alphabetical index are followed by bracketed notations. These notations provide additional clarifying information about the entry such as composition, modifications to the agents (e.g., thickened, dusty, binary), or a note for historical context. For example, "White Star" was a gas blend that was employed by the British in World War I consisting of 50% phosgene and 50% chlorine. The entry appears as ... 

We use the bracket notation of Dirac, following standard practice. The ket J,M> corresponding to (A.32) is also called a basis state. 

Figure 4. Radical chain propagation steps for the pyrolysis of silica-immobilized I, 3-diphenylpropane. Bracket notation denotes species exists in both gas-phase and surface-attached forms.

The bracket notation < > indicates integration over the space and spin coordinates of the electrons. 

The Hermitian property (1.13) expressed in bracket notation and in matrix-element notation is... 

The last line of the text output contains the predicted MFE structure in bracket notation and its free energy in kcal mol-1. A dot in the bracket notation represents an unpaired position, and a base pair (i, j) is represented by a pair of matching parentheses at positions i and /. 

The last two lines are new, compared to the text output without the -p option and are a rough measure for the well-definedness of the MFE structure. The line before the last line shows a condensed representation of the pair probabilities, similar to the bracket notation, followed by the ensemble free energy in kcal mol-1. The structure string contains additional symbols coding for the pairing tendency of that position. 

Inverse folding can be viewed as an optimization problem that can be treated with simple heuristics. This is what the RNAinverse program does for you. Input for the RNAinverse program consists of an RNA secondary structure (the target) in bracket notation (on the first line), optionally followed by a sequence to be used as the starting point of the optimization (otherwise a random start sequence is used). 

The double bracket notation in equation (3) implies the integration over the electronic configuration space By definition, i) f is the ath nonadiabatic... 

Finally, we adopt a notation involving conditional averages to express several of the important results. This notation is standard in other fields (Resnick, 2001), not without precedent in statistical mechanics (Febowitz et al, 1967), and particularly useful here. The joint probability P A, B) of events A and B may be expressed as P A, B) = P A B)P B) where P B) is the marginal distribution, and P A B) is the distribution of A conditional on B, provided that P B) 0. The expectation of A conditional on B is A B, the expectation of A evaluated with the distribution P(A B) for specified B. In many texts (Resnick, 2001), that object is denoted as E(A B) but the bracket notation for average is firmly established in the present subject so we follow that precedent despite the widespread recognition of a notation (A B) for a different object in quantum mechanics texts. 

Of particular importance to the forthcoming development is the statement of the change induced in a property A(p, q) by the generator of an infinitesimal canonical transformation G as expressed in the Poisson bracket notation. The change in the property A obtained as a result of an infinitesimal canonical transformation of the coordinates is given by... 

The bracket notation introduced by Dirac for the matrix element... 

Frequently, the stellar and solar abundance of an element may be compared using the square-bracket notation , i.e., [E/H] = loge(E)star — loge(E)sun- For example, we may speak of Galactic metal-poor stars as those with [Fe/H] < — 1, that is stars with less than one-tenth the Fe abundance of the Sun. The notation is readily extended to a ratio of two elements Ei and E2 for example, a star with [Si/Fe] = +0.5 has a Si/Fe abundance ratio that is three times that of the Sun. Two notes of caution (i) when [X] is quoted check that the Sun was adopted as the reference object, and (ii) A few authors may define [X] quite differently ... 

The square brackets are used to denote a matrix. The are the elements of the matrix. They may be real or complex numbers. Quite often the elements of a matrix will be functions or operators. The matrix [yl] above has p rows and m columns and is said to be a p X m matrix, alternatively, the matrix [ /11 is said to be of order p X m. (It is common to use the letter n to denote the number of rows a matrix has. To avoid confusion with the rest of this book, we have reserved the letter n for denoting the number of components in a mixture.) The order of the matrices we will encounter in this book usually is obvious from the context but, in the event that it becomes necessary to distinguish between matrices of different order, this will be done by appending a subscript to the bracket notation. Thus, the p Xm matrix [A] would be written... 

From the point of view of a computational chemist, one of the most appreciated strengths of the polarization propagator approach is that, although being generally applicable to many fields in physics, it also delivers efficient, computationally tractable formulas for specific applications. Today we see implementations of the theory for virtually all standard electronic structure methods in quantum chemistry, and the implementations include both linear and nonlinear response functions. The double-bracket notation is the most commonly used one in the literature, and, in analogy with Eq. (5), the response functions are defined by the expansion... 

In BK square bracket notation, the first-order zpva correction consists of two terms, i.e. = [P]°a [pji-o. The term [P] ° contains the second derivatives... 

NOTE This appendix is reprinted from U.S. Army (2001c). Boldfaced bracketed notations in the column headed Event Control Number indicate that external investigations were conducted. Legend CDC, Centers for Disease Control and Prevention DAIG. Department of the Army Inspector General DA Safety, Department of the Army Office of Safety DOD. Department of Defense PMCD, Program Manager for Chemical Demilitarization UTAH DEQ, Department of Environmental Quality, State of Utah. 

Bracket notation it consists of two lines. The first line contains the sequence. The second line contains the set of associated pairings encoded by brackets and dots. A basepair between base i and j is represented by a ( at position i and a ) ... 

In the lower part of the RNAz output the predicted structures are explicitly seen for the sequences. There will be structure predictions for each single sequence and a consensus structure prediction for the whole alignment. The predicted structures are given below the sequences in a dot-bracket notation. Each basepair in the secondary structure is indicated by a pair of brackets ( and ) . Unpaired bases are shown as dot Next to the structure the... 

The output contains a consensus sequence and the consensus structure in bracket notation. The bracket notation is convenient string representation for secondary structures. It encodes the structure as a string of dots and matching... 

Here the bracket notation has been used in the sense... 

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