Experiments, quantitative

A general measurement protocol consists of collecting n noisy samples by changing the value of the explanatory variables. To a vector x then corresponds a vector of measurements y in each voxel of the image. This section presents several strategies for choosing x in order to optimize the quality of estimation a priori. 

To optimize or compare measurement protocols, an index of quality must be quantified for each estimated parameter. For a parameter taken separately, the performance of estimation is a function of both the variance (or precision) and the bias (or accuracy) of the estimate. The covariance matrix of each unbiased estimator 0 checks the inequality of Frechet-Cramer-Rao 24 

Provided the estimator is unbiased, this expression indicates that the diagonal elements of the inverse information matrix, the Cramer-Rao bound (CRB), give the highest attainable variance of the unknown parameters. For zero-mean noise and a correct model r, no difference is expected between the values of the estimates and the true values. These minimal variances depend on the pdf of the signal, because the Fisher information matrix I is given by 

Let us consider the case of n magnitude samples. In this case the likelihood function is equal to the product of the Rice pdf, which yields the following expression of the (i, j)th entry of the information matrix 32 

This example has been chosen to show that it is possible to calculate the optimal performance of a measurement protocol in conditions close to those in the field. For a set value of 6 (actually the unknown parameter) it is then possible to seek the value of x that minimizes CRB for a set number n. This optimization is often impossible to carry out analytically when n 2 and/or when matrix I has a complex 

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The results of one of Lavoisier s quantitative experiments are shown in Table A the data are taken directly from Lavoisier. If you add up the masses of reactants and products (expressed in arbitrary units), you find them to be the same, 510. As Lavoisier put it, "In all of the operations of men and nature, nothing is created. An equal quantity of matter exists before and after the experiment."... 

Combustion has a very long history. From antiquity up to the middle ages, fire along with earth, water, and air was considered to be one of the four basic elements in the universe. However, with the work of Antoine Lavoisier, one of the initiators of the Chemical Revolution and discoverer of the Law of Conservation of Mass (1785), its importance was reduced. In 1775-1777, Lavoisier was the first to postulate that the key to combustion was oxygen. He realized that the newly isolated constituent of air (Joseph Priestley in England and Carl Scheele in Sweden, 1772-1774) was an element he then named it and formulated a new definition of combustion, as the process of chemical reactions with oxygen. In precise, quantitative experiments he laid the foundations for the new theory, which gained wide acceptance over a relatively short period. 

The mathematical machinery of thermod3mamics allows this qualitative statement to be expressed quantitatively. Experiments and theory show that the molar entropy of a gas or solute varies logarithmically with concentration... 

The reactions that occur in a mixture of NO2 and N2 O4 lead to changes in concentrations. The concentration of NO2 can be determined experimentally, because NO2 is orange and N2 O4 is colorless. The color intensity of the gas mixture is proportional to the concentration of NO2. Figure 16-2 summarizes the results of quantitative experiments on the NO2 /N2 O4 system. The data show that the concentrations of both gases level off to constant values as the reaction reaches equilibrium. If the reaction to form N2 O4 went to completion, the concentration of NO2 would drop to zero. If the reaction to decompose N2 O4 went to completion, the concentration of N2 O4 would drop to zero. Instead, these reactions reach equilibrium when substantial amounts of both gases are present. 

Probably the earliest quantitative experiments on what are now known as cationic polymerisations were made by Gwyn Williams (1938) with styrene and stannic chloride, and by the early 1940s the general belief had become established that in reactions initiated by metal halides the active species is a cation. It appears that Evans and Meadows [3] were the first to state specifically that in hydrocarbon solvents the propagating cations must be paired with the anions and Plesch [4] made the first attempt at calculating the dissociation constant, KD, for an ion-pair comprising a growing cation in a hydrocarbon solvent ... 

It is difficult because the definitive answers must come from quantitative experiments with extremely pure materials under the most rigorous conditions, and such are very rare. It is complicated because of the great diversity of behaviour which is shown by different metal halides and monomers, and because we are still very ignorant of the chemistry of metal halides. 

Based on the accessibility of high-quality experimental information, we now focus on aspects of model interrogation and analysis. The question how cells actually control and distribute their flux under different conditions requires a mathematical and formal approach to metabolic regulation. The knowledge obtained by quantitative experiments must be, in the sense of Section II.B, encoded into a mathematical system, scrutinized utilizing the tools of formal analysis, and eventually decoded back into predictions about the natural system. 

In 1959 and 1960, Challa published the first results of quantitative experiments on the poly condensation equilibrium in PET [22, 41, 42], He determined the polycondensation equilibrium constant K at different temperatures and average degrees of polycondensation and found that this parameter depends only slightly on temperature, but increases significantly with increasing degree of polycondensation. The monomer BHET was found not to follow the principle of equal reactivity. 

When the student of chemistry desires to effect the transformation of one definite substance into another, he is told to determine, by quantitative experiments, what are the elements, and what the quantities of these elements, which compose the compound which he proposes to change, and the compound into which he proposes to change it and he is given working definitions of the words element and compound. If the compound he desires to produce is found to be composed of elements different from those which form the compound wherewith his operations begin, he is directed to bring about a reaction, or a series of reactions, between the compound which is to be changed, and some other collocation of elements the composition of which is known to be such that it can supply the new elements which are needed for the production of the new compound. 

Leiter, J., and M. J. Shear, Quantitative Experiments on the Production of Subcutaneous Tumors in Strain A. Mice with Marginal Doses of 3,4-Benzpyrene, J. Natl. Cancer Inst., 3, 455-477 (1943). 

Quantitative experiments have been performed to find the stoichiometry of oxygen complexes on surfaces. Oxygen also adsorbs reversibly on a Fe(II) porphyrin attached to the imidazole groups of a silica gel that is adequately treated to give a 1 1 stoichiometry of the Fe-02 adduct (320). 

Quantitative experiments by Daniel Siebert, Jonathan Ott, myself, and others have since confirmed the arithmetic. 

Quantitative experiments have shown that half of its nitrogen appears as ammonia. If the hydrolysis is carried out in dilute sulfuric acid solution, the formaldehyde is oxidized by the nitric acid and nitrous acid is formed. 

Whether or not branching occurs at an intermediate, its existence may be demonstrated by running the reaction in the presence of a reagent designed to intercept that intermediate to yield a new and characteristic product. Observations at a qualitative level can be extremely informative (e.g. formation of cyclo-adducts when aryl halides are treated with a strong base in the presence of conjugated dienes to trap a benzyne intermediate) but even more information maybe obtained from quantitative experiments, especially when product analyses are coupled with rate measurements. 

This reaction, reported briefly in 1912 (2 ), was described in detail later (20). The evolution of carbon dioxide during the reaction was attributed to the carboxyl group of the amino acid after very careful quantitative experiments. 

Rene-Joachim-Henri Dutrochet (1776-1847), French physiologist, conducts the first quantitative experiments on osmosis. He determines that the pressures involved during the diffusion of solutions are proportional to the solution concentrations. He is also the first to observe the motion of particles suspended in a liquid, later called Brownian motion. 

Quantitative experiments have indicated that the F-center EPR signal intensity increased and the Zr3+ signal intensity decreased with the increase in S [50]. Fig. 8.6 shows some graphs constructed from the data published in [50]. 

Careful qualitative and quantitative experiments, such as those performed by Cavendish, did make additional discriminations between gases so that, for example, Cavendish clearly understood the difference between different inflammable airs. It is now time to return to Cavendish. 

Preparations are indicated by P and number and Quantitative Experiments by E and number. 

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