Information-Theoretic Background

The following lower bounds are primarily proved on entropies of random variables, such as the random variable of the secret key generated with a probabilistic key-generation protocol. This automatically yields lower bounds on the length of the values of these random variables (such as individual secret keys), because a random variable with the entropy I cannot be coded with less than / bits on average. 

For the formal theorems and proofs, it is useful if the reader is familiar with elementary information theory (see [Shan48] and [Gall68, Sections 2.2 and 2.3]). The most important notions are briefly repeated in the notation of [Gall68]. It is assumed that a common probability space is given where all the random variables are defined. Capital letters denote random variables and small letters the corresponding vadues, and terms like P(x) are abbreviations for probabilities like P(X = x). The joint random variable of X and Y is written as X, Y. The entropy of a random variable X is 

This does not mean that bounds on time complexity would not be interesting. However, on the one hand, no approach at proving non-trivial ones seems to be known, and on the other hand, efficiency differences between existing signature schemes with different degrees of security primarily concern communication and storage complexity. 

Furthermore, the conditional mutual information between X and Y when Z is known is defined as 

Moreover, if F is a function/of X, the conditional entropy of F given X is zero  

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This chapter presents new information about the physical properties of humic acid fractions from the Okefenokee Swamp, Georgia. Specialized techniques of fluorescence depolarization spectroscopy and phase-shift fluorometry allow the nondestructive determination of molar volume and shape in aqueous solutions. The techniques also provide sufficient data to make a reliable estimate of the number of different fluorophores in the molecule their respective excitation and emission spectra, and their phase-resolved emission spectra. These measurements are possible even in instances where two fluorophores have nearly identical emission specta. The general theoretical background of each method is presented first, followed by the specific results of our measurements. Parts of the theoretical treatment of depolarization and phase-shift fluorometry given here are more fully expanded upon in (5,9-ll). Recent work and reviews of these techniques are given by Warner and McGown (72). 

Hence, for modern FRET and FLIM techniques in Molecular Biology and Biochemistry it is important to keep the enthusiasm for the in situ technique, yielding unprecedented rich information on molecular states in live cells, and to keep the advantages of easy labeling techniques, modern microscopes and automated data processing. However, we need to educate the new generations of FRET scientists in the theoretical background of the technique, how it should be done correctly, and what the sources of errors are. Only then it will be clear that FRET-(FLIM) is a very direct, robust, extremely sensitive, and reliable technique. 

This series of books is to give the newcomer to physical NMR spectroscopy the necessary information, the theoretical background and the practice to acquire NMR spectra, to process the measured raw data from modern routine homo- and heteronuclear ID and 2D NMR experiments, to evaluate NMR parameters, to generate and exploit dedicated data bases and finally to establish molecular structures. 

IH. A BRIEF SURVEY OF THE REQUISITE EXPERIMENTAL INFORMATION AND THEORETICAL BACKGROUND... 

The determination of kLa from an instantaneous reaction is rather complex and the experimental procedure complicated, requiring an extensive knowledge of the theoretical background. Since it is not within the scope of this book to go into the necessary details, the basic experimental procedure is summarized in Figure 3-6, and only a few remarks shall be made here. For complete information the reader is referred to the original literature in which the nonsteady state method has been applied in ozonation experiments (e. g. Beltran and Gonzalez, 1991 Beltran et al., 1992 a Beltran and Alvarez, 1996) and the basics (e. g. Levenspiel and Godfrey, 1974 Charpentier, 1981),... 

It is well known that insertion of the above effective coefficients Se and Pe or De = Pe/Se into Eqs. (2) or (3) respectively, does not lead to the correct description of transient diffusion. However, the behaviour of the ideal Fickian system defined by Se and Pe or De constitutes a useful standard of reference. Given the appropriate theoretical background, one may then deduce information about S(X), DT(X) from the nature and magnitude of the deviation of suitable observed kinetic parameters from the calculated Fickian values. 

The main objective of this review article is to demonstrate the analytical capability of SFM techniques in areas of research where SFM gives the most unique and valuable information not accessible by other methods. Complementary to other reviews [65-69], we will focus on recent developments in SFM instrumentation which are particularly useful for polymer systems (Sect. 2) on quantitative characterisation of material properties and structure manipulation on the nanometer scale (Sect. 3) and on visualisation and probing of single macromolecules (Sect. 4). The interested reader can find the theoretical background as well as instrumentation of SFM in text books [52-54,70-72]. 

Time-resolved emission spectroscopy is gaining importance in the study of various chemical aspects of luminescent lanthanide and actinide ions in solution. Here, the author describes the theoretical background of this analytical technique and discusses potential applications. Changes in the solution composition and/or in the metal-ion inner coordination sphere induce modifications of the spectroscopic properties of the luminescent species. Both time-resolved spectra and luminescence decays convey useful information. Several models, which are commonly used to extract physico-chemical information from the spectroscopic data, are presented and critically compared. Applications of time-resolved emission spectroscopy are numerous and range from the characterization of the... 

AEP, CFP, and RMT stem from the same theoretical background. Whereas the AEP and the CFP can now be regarded as being well-tried procedures, the RMT, especially when applied to the field of molecular dynamics in the liquid state, has still to be considered as a first attempt at settling in a unitary picture the huge amount of information provided by computer experiment. The ultimate reason for the presence in this volume of some chapters on the subject of computer experiments is to underline the program of this theoretical approach, a rigorous version of which is not yet available, so as to stimulate further research toward the attainment of a reliable and comprehensive formulation. 

Together with Mans Ehrenberg who joined me as my first doctoral student we had cleared up the theoretical background of rotational relaxation after anisotropic excitation of the groundstate with a pulse of polarized light [5], and we were involved in the first experiments using the distribution of photon arrival times after a pulsed excitation. The limitation of this approach in our eyes was the fact that the rotational anisotropy decay was linked directly to the life time of the excited state and no information on rotational motion was available after the excited state had vanished. 

The theoretical background which will be needed to calculate the excited state distortions from electronic and Raman spectra is discussed in this section. We will use the time-dependent theory because it provides both a powerful quantitative calculational method and an intuitive physical picture [42,46-50]. The method shows in a simple way the inter-relationship between Raman and electronic spectroscopy. It demonstrates that the intensity of a peak in a resonance Raman spectrum provides detailed information about the displacement of the excited state potential surface along the normal mode giving rise to the peak [42,48]. It can also be used to calculate distortions from the intensities of vibronic peaks in electronic spectra [49]. For harmonic oscillators, the time-dependent theory is mathematically equivalent to the familiar Franck-Condon calculation [48]. 

A thorough discussion of the molecular and physical properties, with the appropriate theoretical background, is beyond the scope of this chapter. 11)0 information is also extensively available in literature. Instead, the discussion will focus on properties that can... 

The intention of the author was to provide information on pyrolysis for a wide range of readers, including chemists working in the field of synthetic polymers as well as for those applying pyrolysis coupled with specific analytical instrumentation as an analytical tool. Some theoretical background for the understanding of polymer structure using analytical pyrolysis is also discussed. The book is mainly intended to be useful for practical applications of analytical pyrolysis in polymer identification and characterization. 

The basic approach of this edition is little changed from the first the emphasis is still on the review of methods and applications which are most useful for quantitative, analytical determination of ions in a wide variety of matrices. An ultimate practitioner of ion chromatography, the author has added a substantial amount of data from his own applications development work. The theoretical background description on various subjects of ion determination is short but informative, and is written so that a novice in the field will not only read and understand it, but also enjoy it. Experts in the field, on the other hand, will undoubtedly find Dr. Weiss s new text a useful reference for many applications and practical problems faced by an analytical chemist, ranging from the field of water purity analysis to the complex task of carbohydrate analysis of glycoproteins. 

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