Early conjectures

The exact values of the exponents v and co corresponding to polymers interacting in a two-dimensional space can be considered as known. They have been found by means of field theory. The formulae given by Cardy and Hamber54 and confirmed by Nienhuis (1982)52 led to the following results 

The results found by Cardy and Hamber, and by Nienhuis, apply more generally to models of the Landau-Ginzburg type with n-component fields. In the general case, the exponents y, v, and co are given in terms of n (with — 2 n 2) in the parametric form 

In order to justify the conjecture made by Cardy and Hamber, Nienhuis used a cascade of models (including an unsolved six-vertex model)4 and equivalences that are more or less exact finally, he came down to a two-dimensional Coulomb gas. This gas is made of positively and negatively charged particles in interaction, the interaction potential being proportional to Inr where r is the distance between two charges. Then, Nienhuis could apply to this system approximate renormalization techniques which enabled him to predict the critical properties of the system. 

These subtle and complicated arguments are not rigorous and will not be reproduced here. One reason is that the same results can be found by using a very promising method based on properties of conformal invariance. We shall come back to this question in Section 4.2. In what follows we only show how Nienhuis began to tackle the problem. 

The starting model is the 0(n) model defined as follows. We associate random vectors M with the sites M of a hexagonal lattice (honeycomb lattice, see Fig. 12.14). Each vector SM has n components Si, which have the following stochastic properties 

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The most characteristic aspect of the critical point problem is that the three phenomena, cyclization, excluded volume effects, and dimension, intimately interacting with each other, spontaneously appear at the critical point. At the beginning, it was thought that cyclization would make little contribution to such an important question that has remained unsolved for so long in physical science. The author s early conjecture was wrong. As we have seen in the text, cyclization plays a central role in the location of the critical point. For the percolation model, dimension is almost equivalent to cyclization (Sects. 4 and 5) even excluded volume effects seem to manifest themselves as an element of cyclization (Sects. 6 and 7), while dimensionality is in close conjunction with excluded volume effects (Sect. 7). In real gelations, the three effects are deeply connected with one another. 

An early conjecture of Dislich in this direction is noteworthy. The extraordinary homogeneity of the final ceramic product obtained by the sol-gel hydrolysis of a mixtnre of alkoxides of various metals in alcoholic solution, led him to suggest that in addition to the advantage of much more efficient mixing at the molecular level in the initial precnrsor solntion, some sort of complexation reactions must be occurring amongst the alkoxides of various metals in the initial stages ... 

The nature of the bonds between cellulosic fibres in paper has been the subject of some controversy over many years. The early and now largely discredited view was that paper derived its strength merely from mechanical entanglement of the fibres. However, experiments in which paper is formed from non-aqueous solvents produce sheets with very poor strength properties and have thus tended to disprove this conjecture. In the mid-1950s deuteration experiments were carried out which demonstrated that of the order of 0.4-2% of all hydroxy groups are additionally bonded in paper as compared with the unbonded fibres. This observation led to the view that... 

In the early thirties of the last century Baade and Zwicky conjectured in their studies of supernova explosions that supemovae represent a transition from ordinary stars to compact objects, whose size is an order of magnitude smaller than the size of a white dwarf. At that time it was already known that the atomic nucleus consists of neutrons and it was clear that the density of the remnant objects must be of the same order as the nuclear density. Baade and Zwicky predicted that a supernova explosions will result in objects composed of closely packed neutrons (neutron stars). Prior to the beginning of the second World War (1939) a number of theoretical works by Landau, Oppenheimer, Volkoff and Snider showed, that indeed objects could exist with sizes about 10 km and masses about a solar mass. The density in these objects is about the nuclear saturation density and they basically consist of neutrons with a small amount of protons and electrons. The studies of neutron stars were subsequently stopped most likely due to the engagement of the nuclear scientists in the development of the nuclear bomb both in the West and the East. 

Early attempts at observing electron transfer in metalloproteins utilized redox-active metal complexes as external partners. The reactions were usually second-order and approaches based on the Marcus expression allowed, for example, conjectures as to the character and accessibility of the metal site. xhe agreement of the observed and calculated rate constants for cytochrome c reactions for example is particularly good, even ignoring work terms. The observations of deviation from second-order kinetics ( saturation kinetics) allowed the dissection of the observed rate constant into the components, namely adduct stability and first-order electron transfer rate constant (see however Sec. 1.6.4). Now it was a little easier to comment on the possible site of attack on the proteins, particularly when a number of modifications of the proteins became available. 

Table 10.2 lists upper concentration limits for Henry s law behavior of various trace elements in silicate crystals for specific P and T conditions. In some cases,th-ese limits are only approximate or conjectural. The table is not exhaustive, because it reflects the state of the art in the early 1980s and is subject to substantial modifications in response to the advance of experimental knowledge since that time. 

The atmosphere of modern Earth is thought to be very different from that of early Earth. Scientists conjecture that Earth s first atmosphere consisted of carbon dioxide, water vapor, nitrogen, and hydrogen sulfide, with trace amounts of ammonia and methane. The gases in the atmosphere are thought to have been released from the interior of the planet by volcanic eruptions. At this early... 

If, on the other hand, he means only the separation of air from water by boiling, as illustrated in the latter of the above sentences, then it may be conjectured that lie only observes the conversion of water into vapor (air) in. boiling. In no case is there any justification for Iloefer s conclusion. Paracelsus and Basilius Valentinus may therefore both be eliminated as early observers of the formation of the gas now known as hydrogen. 

It is a matter of conjecture as to whether sunlight was involved in the buildup of early organic molecules which eventually formed DNA, RNA and proteins. Photosynthetic bacteria may well have been the very first independent life forms, and from the time of the growth of green plants the atmosphere must have become gradually richer in oxygen, since it appears as a byproduct in the major process of photosynthesis. It is generally accepted that early life developed in the oceans, and it may be surmised that its eventual development on land was made possible by the formation of the protective ozone layer in the upper atmosphere. 

The absence of a formal nomenclature al this juncture is accompanied by a somewhat fuzzy chronology pertaining to the discovery and early research on ihe fullerenes. However, the isolation and confirmation of the CHI all-carbon molecule sans any dangling bonds, as first conjectured in 1985. was pivotal to subsequent research. 

While the early work on molten NH4CI gave only some qualitative hints that the effective critical behavior of ionic fluids may be different from that of nonionic fluids, the possibility of apparent mean-field behavior has been substantiated in precise studies of two- and multicomponent ionic fluids. Crossover to mean-field criticality far away from Tc seems now well-established for several systems. Examples are liquid-liquid demixings in binary systems such as Bu4NPic + alcohols and Na + NH3, liquid-liquid demixings in ternary systems of the type salt + water + organic solvent, and liquid-vapor transitions in aqueous solutions of NaCl. On the other hand, Pitzer s conjecture that the asymptotic behavior itself might be mean-field-like has not been confirmed. 

A mixture known as black powder revolutionized the art of warfare whenever it was applied to the propulsion of missiles. Black powder is a mixture of potassium nitrate (saltpeter), charcoal, and sulfur in varying proportions, granulation, and purity. A typical composition of a modern black powder is saltpeter 75%, charcoal 15%, and sulfur 10%.7 A mixture of saltpeter, charcoal, and sulfur with other ingredients was used in China and India in the eleventh century for incendiary and pyrotechnic purposes long before true black powder was invented.8 History often deals in conjecture and opinion and it is not known for certain when and by whom black powder was invented, or when and by whom it was applied to the propulsion of a missile from a firearm. The composition of black powder was first recorded by English Franciscan monk Roger Bacon in 1249, but he did not apply it to the propulsion of a missile from a firearm. This use of black powder is usually credited to a German Franciscan monk Berthold Schwartz in the early fourteenth century.9... 

A hot fluid model would be highly desirable for applications in astrophysics. As we have already mentioned, the formation of RES in the primordial plasma could be an important source of large-scale nonuniformities in density and temperature, which seeded the formation of galaxies and clusters of galaxies [4], In particular, it is conjectured that in the early universe matter was present in the form of a mixture of electrons, positrons and photons in thermal equilibrium at a temperature above me2. It is evident that the propagation of relativistic EM waves in such peculiar environment should be addressed in the framework of a hot-plasma model. 

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