Common factoring

After substitution of the leading terms of the expanded variables into the model equations and equating coefficients of equal powers of e from their sides, they are divided by common factors to obtain the following set ... 

Since each term in the numerator and the denominator contains the factor (1 - p), this common factor cancels out ... 

Polyesters are eneountered in many forms. They are important as laminating resins, moulding compositions, fibres, films, surface coating resins, rubbers and plasticisers. The common factor in these widely different materials is that they all contain a number of ester linkages in the main chain. (There are also a number of polymers such as poly(vinyl acetate) which contain a number of ester groups in side chains but these are not generally considered within the term polyester resins.)... 

The first requirement must be to specify carefully the operational requirements of the item to be produced and of the material(s) to be used in its construction. This is seldom easy and is usually the most difficult part of the selection process. The most common factors to be considered are ... 

The collection efficiency of cyclones varies as a function of particle size and cyclone design. Cyclone efficiency generally increases with (1) particle size and/or density, (2) inlet duct velocity, (3) cyclone body length, (4) number of gas revolutions in the cyclone, (5) ratio of cyclone body diameter to gas exit diameter, (6) dust loading, and (7) smoothness of the cyclone inner wall. Cyclone efficiency will decrease with increases in (1) gas viscosity, (2) body diameter, (3) gas exit diameter, (4) gas inlet duct area, and (5) gas density. A common factor contributing to decreased control efficiencies in cyclones is leakage of air into the dust outlet (EPA, 1998). 

Note that a common factor in incidents 8.1.1 through 8.1.3 was a failure by those concerned to understand the properties of the materials and equipment. 

In many ways factories, at least, ARE like they used to be. This is not surprising, as human nature is a common factor. We have better equipment but may be just as likely as in the past to cut corners when we design, construct, operate, test, and maintain it, perhaps more likely as there are fewer of us to keep our eyes open as we go round the plant and to follow up unusual observations. We have access to more knowledge than our parents and grandparents, but are we any more thorough and reliable ... 

As a point P(x,y) on a curve moves away from the region of the origin (Figure 1-36), the distance between P and some fixed line may tend to zero. If so, the line is called an asymptote of the curve. If N(x) and D(x) are polynomials with no common factor, and... 

Catalytic conversion processes include naphtha catalytic reforming, catalytic cracking, hydrocracking, hydrodealkylation, isomerization, alkylation, and polymerization. In these processes, one or more catalyst is used. A common factor among these processes is that most of the reactions are initiated hy an acid-type catalyst that promotes carhonium ion formation. 

The type of plant which would be included in the above generic description will include many items that, because of difference in size, function and appearance, will appear to be unrelated. They do, however, have one common factor in that all operate to some degree under pressure. 

Industrial effluents are a particularly difficult problem to discuss in general terms their nature is very diverse, possible methods of treatment vary correspondingly and their acceptability depends as much on the receiving body as on their flow and contents. There are, however, some common factors which are worth mentioning. 

Review of Solutions in General. In the discussion of these various examples we have noticed at extreme dilution the prevalence of the term — In Xb, or alternatively — In yB. The origin of this common factor in many different types of solutions can be shown, as we might suspect, to be of a fundamental nature. For this purpose let us make the familiar comparison between a dilute solution and a gas. Since the nineteenth century it has been recognized that the behavior of any solute in extremely dilute solution is, in some ways, similar to that of a gas at low pressure. Now when a vessel of volume v contains n particles of a perfect gas at a lixed temperature, the value of the entropy depends on the number of particles per unit volume, n/v. In fact, when an additional number of particles is introduced into the vessel, the increment in the entropy, per particle added, is of the form... 

Why are these conditions important Do they have something in common Yes, there is the common factor that a candle does not operate well in a draft. The conditions are important because they influence ihe result of the experi-... 

Equation (7) is also a chemical equation—again atoms are conserved. It is just as correct an expression for the burning of magnesium as is (6). To show this, we can multiply (7) by 2 to obtain equation (6). We can always multiply all the coefficients by a common factor or divide by a common factor and obtain equally valid equations. 

If sulfuric acid, H2SO4, is added to an aqueous solution of formic acid, carbon monoxide bubbles out rapidly. This also occurs if phosphoric add, HjPO, is added instead. The common factor is that both of these acids release hydrogen ions, H+. Yet, careful analysis shows that the concentration of hydrogen ion is constant during the rapid decomposition of formic acid. Evidently, hydrogen ion acts as a catalyst in the decomposition of formic acid. 

What is the common factor that makes these different substances behave in the same ways In water they all form conducting solutions we conclude that they all form ions in water. Each substance contains hydrogen and each reacts with zinc metal to produce hydrogen gas. Perhaps all of these aqueous solutions contain the same ion and this ion accounts for the formation of Hfg). It is reasonable to propose that the common ion is H+(aq). We postulate a substance has the properties of an add if it can release hydrogen ions. 

Using the same argument we used for acids, we can seek a common factor that accounts for the similarities of bases. Because of the electrical conductivity, we might seek an ion. Because of the ability to counteract the properties of acids, we ought to seek an ion which can remove the hydrogen ion, H+faq), since hydrogen ion accounts for the properties of adds. 

Some of the common factors that control the accuracy of these direct methods are incomplete removal of water, thermal decomposition, volatility of nonaqueous components, and certain side reactions, such as oxidation and nonspecificity of the Fischer reagent. 

In methods involving image formation there is a common factor. Radiation normally propagated in rectilinear fashion, such as light or an electron beam, is interrupted by the particles under examination, and the pattern of the interruption can be observed in different ways, eg, optically or photo-electrically... 

A simple expression for the signal-to-noise ratio (SNR) of a measurement of visibility amplitude involves several parameters relating to interferometer and source properties. The formula presented here provides the fundamental sensitivity limit. Contrast loss arising from instrumental jitter and seeing are summarised in a common factor system Strehl , which is the ratio of the number of photons which can be used for a coherent measurement to the... 

If some results from Step 2 are far from whole numbers, multiply through by a common factor that makes all molar amounts close to whole numbers. 

Suppose, for clarity, we add a common factor [(V2)jn + (V2)yc] to all four energy levels so the energy difference between them does not change. The values of these energy levels will then become y, yc, Jh, and yH + yc-Since yn is about four times yc, let us also assume that yn = 4 and yc = 1. The populations of the four energy levels will then be 0, 1, 4, and 5. The population difference between the two C spin states before the application of the H polarization transfer pulse corresponds to the lower energy state minus the upper energy state, i.e. 1—0=1 or 5 — 4=1. 

As to the second chief point of this paper, let me finish by restating It. If you want to solve a nitrosamlne contamination problem, learn everything you can about the chemistry of your product and the process by which it was made, then learn everything you can about the mechanisms of nitrosamlne formation. Sifting through the two sets of Information for common factors should provide the simplest, safest, most systematic means by which you can understand, control and predict environmental contamination by carcinogenic nitrosamlnes. 

The APFl gene product, a common factor of unknown function which increases the activity of amino acid permeases... 

Fig. 31.2. Geometrical example of the duality of data space and the concept of a common factor space, (a) Representation of n rows (circles) of a data table X in a space Sf spanned by p columns. The pattern P" is shown in the form of an equiprobabi lity ellipse. The latent vectors V define the orientations of the principal axes of inertia of the row-pattern, (b) Representation of p columns (squares) of a data table X in a space y spanned by n rows. The pattern / is shown in the form of an equiprobability ellipse. The latent vectors U define the orientations of the principal axes of inertia of the column-pattern, (c) Result of rotation of the original column-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the score matrix S and the geometric representation is called a score plot, (d) Result of rotation of the original row-space S toward the factor-space S spanned by r latent vectors. The original data table X is transformed into the loading table L and the geometric representation is referred to as a loading plot, (e) Superposition of the score and loading plot into a biplot.

Fig. 31.3. (a,b) Reproduction of distances D and angular distances 0 in a score plot (a = 1) or loading plot (p = 1) in the common factor-space (c,d) Unipolar axis through the representation of a row or column and through the origin 0 of space. Reproduction of the data X is obtained by perpendicular projection of the column- or row-pattern upon the unipolar axis (a + P = 1). (e,0 Bipolar axis through the representation of two rows or two columns. Reproduction of differences (contrasts) in the data X is obtained by perpendicular projection of the column- or row-pattern upon the bipolar axis (a + P = 1). 

Obviously minimization of SML(k) in the above equation does not require the prior knowledge of the common factor ae. Therefore, under these conditions the ML estimation is equivalent to simple LS estimation (Q,=I). 

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