discrete mathematics for computer scientists solution manual pdf

Ngày tải lên: 14/08/2014, 04:21

... original formula for q. Recall that our proof of the formula we had in Exercise 1.4-5 did not explain why the product of three factorials appeared in the denominator, it simply proved the formula ... distinct elements. There are n choices for the first number in the list. For each way of choosing the first element, there are n −1choices for the second. For each choice of the first two elements, ... the product in the denominator of the formula in Exercise 1.4-5 for the number of labellings with three labels is what it is, and could generalize this formula to four or more labels. Equivalence...

Ngày tải lên: 21/02/2014, 09:20

... possible, and so the total score for the game is 1(1 − 1)/2 = 0. Therefore, P(1) is true. Inductive step: Now we must show that P (1), . , P (n − 1) imply P (n) for all n ≥ 2. So assume that P ... 1) for all n ∈ N. By the principle of induction, P (n) is true for all n ∈ N, which proves the claim. This proof would look quite mysterious to anyone not privy to the scratchwork we did beforehand. ... divisibility hold. 1. If a | b, then a | bc for all c. 2. If a | b and b | c, then a | c. 3. If a | b and a | c, then a | sb + tc for all s and t. 4. For all c = 0, a | b if and only if ca | cb. Proof....

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Ngày tải lên: 10/10/2014, 13:39

... acid or dilute basic solution. When adjusting the w w pH of a buffer solution, make sure to wait until the solution reaches equilibrium after adding additional acid or base before measuring the ... sample solutions are filtered and/or centrifuged; the use of a 0.2- or 0.45-µm-pore-size filter is generally recommend for removal of particulates [15]. Filtration as a preventive maintenance tool for ... it will also significantly decrease the peak width (especially for the late eluting analytes), thus increasing the resolution. This resolution increase obtained with a linear gradient often outweights...

Ngày tải lên: 15/12/2013, 12:15

... The experiments for w w pH 2–4 were performed on day 1, and those for w w pH 5–7 were performed on day 2 (≈36 hr after initial prepa- ration).A further investigation was performed by preparing ... requirement for injection precision, sensitivity, standard accuracy (if for an assay method), and retention time of the target analyte. Sometimes, a resolution requirement is added for a critical ... “chromatographer’s toolbox” allows for the development of rugged and reproducible methods. 8.5.6 Case Study 3: Method Development for a Diprotic Basic Compound A case study is presented for the method development...

Ngày tải lên: 15/12/2013, 12:15

... Figure 8-40A for the sample stored in the acidic diluent for 4 days at room temperature shows some degradation compared to Figure 8-40B for the sample stored in the acidic diluent for one day ... Flow rate for all four runs was 0.8 mL/min. determine the resolution map for the active and impurity 11 eluting close to the main component (Figure 8-46). In a similar fashion, a resolution map ... analyte pK a shift) for both basic ionization centers. Therefore this cor- relates with a s s pK a of 2.7 for pyridinal nitrogen (i.e., 3.3 − 0.6 = 2.7) and with a s s pK a of 4.7 for the morpholinal...

Ngày tải lên: 15/12/2013, 12:15

... Similar parameters have been determined for this family of compounds for methanol/water mixtures [80]. Using these parameters for each family of compounds for a particular type of organic, the a s and ... same critical distance (D) needed for ion-pair formation, a solvent with a high dielectric constant such as water (∼80) will be less favorable for ion-pair formation compared to a solvent that ... obtain the acid analyte in its fully neutral form? Step 1. First account for the upward pK a shift for the acidic analyte upon addi- tion of organic. For every 10 v/v% increase in acetonitrile...

Ngày tải lên: 24/12/2013, 18:15

... might be obtained for the basic analyte in its neutral form.The advan- tages of employing chaotropic mobile-phase additives at a pH where the basic analyte is in its fully protonated form provides ... mobile-phase pH adjustment. A chaotropic approach could be used for the separation of very polar basic compounds as a fast screening method for the resolution of closely eluting basic species without resorting ... chromatographic methods for the determination of adsorption and solution isotherms, J. Chro- matogr. 58 (1971), 137–158. 182. H. L. Wang, J. L. Duda, and C. J. Radke, Solution adsorption from...

Ngày tải lên: 24/12/2013, 18:15

... the triangle formed by MTBE, dichloromethane, and chloroform defines the selectivity domain in which the optimum mobile-phase composition will be found. Next, separations are performed with three ... reliable for less polar solvents and solute molecules on alumina or silica stationary phases only. Neither of the models is entirely satisfactory in the forms presented, par- ticularly for predicting ... zirconia for normal phase chromatography, J. Chromatogr. A 691 (1995), 205–212. 22. U. Trüdinger, G. Müller, and K. K. Unger, Porous zirconia and titania as packing materials for high-performance...

Ngày tải lên: 24/12/2013, 18:15

... chro- matographic peaks for a given sample will be hard to predict, and therefore the footprint for fraction collection will be difficult to predict. Experience has shown that it is not uncommon for 5–10 fractions ... solid-phase and solution- phase synthesis of discretes [27–31]. Both solution- phase and solid-phase parallel synthesis permits the production of large numbers as well as large quantities of these discrete ... by Xu et al. was used for assessing the time-course metabolic stability (four time-points in triplicate) for hits identified from screening of lead generation libraries. For each compound, a total...

Ngày tải lên: 22/01/2014, 00:20

... eluent for reversed- phase high performance liquid chromatography, Anal. Commun. 33 (1996), 327–329. 57. R. M. Smith and R. J. Burgess, Superheated water as an eluent for reversed-phase high performance ... reversed-phase high-performance liquid chromato- graphy, J. Chromatogr. A 499 (1990), 79–88. 85. J.A.Whatley, Chiral resolution of protein-kinase inhibitors by reversed-phase high- performance liquid ... Zirconia stationary phases for extreme separations, Anal. Chem. 73 (2001), 598A–607A. 49. J. Li and P.W. Carr,Effect of temperature on the thermodynamic properties, kinetic performance, and stability...

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