Fields of definition

Let k be an algebraically closed field and let X be a closed subvariety of An, defined by equations  

Suppose that we are particularly interested in some subfield ho of k (usually because ko is important for arithmetic reasons, or because ko has an interesting topology). Then if the coefficients of the all lie in ko, it will be very important to make use of this fact it may, for example, pose Diophantine or rationality questions. When this happens one says that X is defined over koLet 

Then if the coefficients of the lie in ko, it follows firstly that A = k Aq where Ao = An ko [X, . Xn] and secondly, if Ro = ko [X, . Xn] /Aq, then the affine ring R = k [Xi. Xn] /A of X is of the form Rq 8 k0 This shows us the scheme-theoretic significance of X being defined over ko . there exists an affine scheme Xq = Spec (Rq) of finite type over ko such that 

Assume that we axe given any prescheme Xo over Aft and that we define X as Xo Xgpec (ko) Spec (k) we shall often write X = X0 xfco A for simplicity. Explicitly, if Xo is the union of open affine sets (Uo)i = Spec ( ), then X is just the union of the affine schemes 

One checks immediately that ( t-t)x = ax TX, i.e., the Galois group of k/ko is acting on the topological space X. In particular, each ax is a homeomorphism of X. In simple cases, this conjugation is exactly what you expect it to be Assume 

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Crystal field theory gives a survey of the effects of electric fields of definite symmetries on an atom in a crystal structure. 

Clearly, the study of CMR (GMR) systems containing rare earths has started and more work is to be expected in the near future. It has been demonstrated that nSR can provide important information on local aspects of the magnetic properties of these substances and in particular on the nature of phase transitions. The repeated appearance of spin freezing is noteworthy. Spin dynamical properties are another field of definite interest wich is open to p.SR It is generally believed that the transport properties of the manganese compounds are coupled to lattice and Mn spin dynamical effects. As of yet, it is not clear how well the findings of p.SR relate to the phenomenon of CMR (or GMR). 

The field of metaHurgy has a unique and frequently very specialized vocabulary. Understanding this language helps to clarify certain concepts and processing steps. A complete dictionary of mining, mineral, and related terms has been compiled (2). The definitions and explanations of key terms foHow. 

It is becoming more and more desirable for the analytical chemist to move away from the laboratory and iato the field via ia-field instmments and remote, poiat of use, measurements. As a result, process analytical chemistry has undergone an offensive thmst ia regard to problem solviag capabihty (77—79). In situ analysis enables the study of key process parameters for the purpose of definition and subsequent optimization. On-line analysis capabihty has already been extended to gc, Ic, ms, and ftir techniques as well as to icp-emission spectroscopy, flow iajection analysis, and near iafrared spectrophotometry (80). 

The first definitive studies of boron hydrides were carried out by Alfred Stock in Germany starting about 1912 (1). Through extensive and now classic synthetic studies, the field of boron hydride chemistry was founded with the isolation of a series of highly reactive, air-sensitive, and volatile compounds of general composition and This accomplishment required the development of basic vacuum line techniques for the... 

The fundamental parameters in the two main methods of achieving ignition are basically the same. Recent advances in the field of combustion have been in the development of mathematical definitions for some of these parameters. For instance, consider the case of ignition achieved by means of an electric spark, where electrical energy released between electrodes results in the formation of a plasma in which the ionized gas acts as a conductor of electricity. The electrical energy Hberated by the spark is given by equation 2 (1), where V = the potential, V 7 = the current. A 0 = the spark duration, s and t = time, s. 

Good heat transfer on the outside of the reactor tube is essential but not sufficient because the heat transfer is limited at low flow rates at the inside film coefficient in the reacting stream. The same holds between catalyst particles and the streaming fluid, as in the case between the fluid and inside tube wall. This is why these reactors frequently exhibit ignition-extinction phenomena and non-reproducibility of results. Laboratory research workers untrained in the field of reactor thermal stability usually observe that the rate is not a continuous function of the temperature, as the Arrhenius relationship predicts, but that a definite minimum temperature is required to start the reaction. This is not a property of the reaction but a characteristic of the given system consisting of a reaction and a particular reactor. 

In spite of numerous advances in the field of detection there are not and never have been any genuinely substance-specific chemical detection reactions. This means that, unlike the spectrometric methods, the methods of detection normally employed in chromatography cannot be employed for an unequivocal identification of compounds, they can only provide more or less definite indications for the characterization of the separated substances. Universal reagents are usually employed for a first analysis of the separation of samples of unknowns. This is then followed by the use of group-specific reagents. The more individual the pieces of information that can be provided from various sources for a presumed substance the more certainly is its presence indicated. However, all this evidence remains indicative it is not a confirmation of identity. 

An important field of study for power plants is that of the combinedplant [ 1 ]. A broad definition of the combined power plant (Fig. 1.5) is one in which a higher (upper or topping) thermodynamic cycle produces power, but part or all of its heat rejection is used in supplying heat to a lower or bottoming cycle. The upper plant is frequently an open circuit gas turbine while the lower plant is a closed circuit steam turbine together they form a combined cycle gas turbine (CCGT) plant. 

An extensive range of definitions in the field of quality management is provided in ISQ 8402 1994 and Appendix A includes over 150 commonly-used terms, less verbose but consistent with the definitions found in ISQ 8402 1994. 

Most of the analytical structure of the dynamics of linear CA systems emerges from their field-theoretic properties specifically, those of finite fields and polynomials over fields. A brief summary of definitions and a few pertinent theorems will be presented (without proofs) to serve as reference for the presentation in subsequent sections. 

Some Number Theory Definitions Let be a field. We recall that any subset K, extension field of 1C. Now let / A[ e) be. some polynomial of po.sitive... 

Analysis of the correlation between concentrations and the intensity of the bands at 699 cm 1 and 645 cm 1, which refer to TaF6 and TaF72 complex ions respectively, enable definition of the predominant presence fields of the two ions. These fields are shown in Fig. 50. 

It is not only in the field of kinetic relations that discrepancies exist. When the catalyst is a protonic acid and the reaction is carried out in dilute solution, the mechanisms describing the contribution of the catalyst are relatively well-known. But in most other cases and particularly when the catalyst is a metal derivative (see Chap. 4) none of the proposed mechanisms can be considered as definitive. 

The use of electrical engineering terminology here is purely for the sake of definiteness. The results obtained apply to a wide variety of problems arising in such diverse fields as mechanical vibrations, acoustics, and, with t replaced by a space parameter, optics. 

Secondly, I wish to counteract anticipated despondency which some of the complexities on the present theoretical scene may perhaps provoke. For this purpose, I wish to invoke the decisive simplicity and definiteness of some of the experimental effects observed within the confines of the above, near ideal systems. This, as I often pointed out elsewhere, is unmatched in the field of crystal growth of simple substances. Complicated as polymers may seem, and subtle as some of the currently relevant theoretical issues, this should not obscure the essential simplicity and reproducibility of the core material. To be specific, the appropriate chains seem to want to fold and know when and how, and it is hardly possible to deflect them from it. Clearly, such purposeful drive towards a predetermined end state should continue to give encouragement to theorists for finding out why Those who are resolved to persevere or those who are newly setting out should find the present review a most welcome source and companion. 

AG type II is most abundant in the heartwood of the genus Larix and occurs as minor, water-soluble components in softwoods. Certain tree parts of western larch (I. occidentalis) were reported to contain up to 35% AG [378]. The polysaccharide is located in the lumen of the tracheids and ray cells. Consequently, it is not a cell-wall component and, by definition, not a true hemicellulose. However, it is commonly classified as such in the field of wood and pulping research. This motivated us to include the larch AG in the review. 

It follows that for a special value of one parameter, the observed value of y is independent of the second parameter. This happens at Ii= a2/ai2 or I2 = -ai/ai2 any of these values determines y= a -aia2/ai2, the so called isoparametrical point. The argument can evidently be extended to more than two independently variable parameters. Experimental evidence is scarce. In the field of extrathermodynamic relationships, i.e., when j and 2 are kinds of a constants, eq. (84) was derived by Miller (237) and the isoparametrical point was called the isokinetic point (170). Most of the available examples originate from this area (9), but it is difficult to attribute to the isoparametrical point a definite value and even to obtain a significant proof that a is different from zero (9, 170). It can happen—probably still more frequently than with the isokinetic temperature—that it is merely a product of extrapolation without any immediate physical meaning. 

At present the situation in the field of inorganic polymeric materials is dominated by polysiloxanes (silicones) [14, 24-27], whose utilization as low temperature elastomers, thermally stable fluids, biomaterials etc., is definitely well established. 

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