On Aperiodic Subtraction Games with Bounded Nim Sequence

Subtraction games are a class of impartial combinatorial games whose positions correspond to nonnegative integers and whose moves correspond to subtracting one of a fixed set of numbers from the current position. Though they are easy to define, sub- traction games have proven difficult to analyze. In particular, few general results about their Sprague-Grundy values are known. In this paper, we construct an example of a subtraction game whose sequence of Sprague-Grundy values is ternary and aperiodic, and we develop a theory that might lead to a generalization of our construction.Comment: 45 page

Spectra of Semidirect Products of Cyclic Groups

The spectrum of a graph is the set of eigenvalues of its adjacency matrix. A group, together with a multiset of elements of the group, gives a Cayley graph, and a semidirect product provides a method of producing new groups. This paper compares the spectra of cyclic groups to those of their semidirect products, when the products exist. It was found that many of the interesting identities that result can be described through number theory, field theory, and representation theory. The main result of this paper gives a formula that can be used to find the spectrum of semidirect products of cyclic groups

The Intuition of Knowing: Its Biological Function and Natural Triggering-Conditions

Over the last hundred years, competing and incompatible positions in relation to basic problems of knowledge and the use of the verb ‘to know’ have multiplied; and the prospect of a consensus solution emerging with respect to any of the problems has not seemed particularly good. We have a Gordian knot. Even so, I suggest that we also have a way to cut it. This will involve identifying why the cognitive mechanism that produces our intuitions of knowing evolved and was maintained (by natural selection), i.e., identifying the ‘teleonomic function’ of that cognitive mechanism. Also, it will involve predicting, on the basis of this teleonomic function, the triggering-conditions of these natural knowledge intuitions. In this thesis, I develop a general theory of the origin, function and triggering-conditions of knowledge intuitions that will allow us to cut that knot. That theory follows basic biological theory (including that which pertains to natural altruism) and also signal detection theory. My theory identifies a number of different circumstances under which the triggering-conditions of knowledge intuitions are different. Strikingly, these different circumstances (and their associated triggering-conditions) map onto the different competing and incompatible epistemological positions to which I referred. This suggests that these positions are all correct within the boundaries of one of the circumstances that my theory identifies; and that the Gordian knot is largely the result of epistemologists claiming universal applicability of a theory that in fact only applies under particular circumstances. We cut the knot by specifying the different circumstances under which each of the different epistemological positions will hold, and the reason we should expect it to hold in just these circumstances, in light of the teleonomic function of knowledge intuitions

Hot corrosion of ceramic engine materials

A number of commercially available SiC and Si3N4 materials were exposed to 1000 C in a high velocity, pressurized burner rig as a simulation of a turbine engine environment. Sodium impurities added to the burner flame resulted in molten Na2SO4 deposition, attack of the SiC and Si4N4 and formation of substantial Na2O-x(SiO2) corrosion product. Room temperature strength of the materials decreased. This was a result of the formation of corrosion pits in SiC, and grain boundary dissolution and pitting in Si3N4. Corrosion regimes for such Si-based ceramics have been predicted using thermodynamics and verified in rig tests of SiO2 coupons. Protective mullite coatings are being investigated as a solution to the corrosion problem for SiC and Si3N4. Limited corrosion occurred to cordierite (Mg2Al4Si5O18) but some cracking of the substrate occurred

Molten salt corrosion of SiC and Si3N4

Industrial systems such as heat engines and heat exchangers involve harsh environments. The structural materials are subjected to high temperatures as well as corrosive gases and condensed phases. Past experience with metal alloys has shown that these condensed phases can be particularly corrosive and are often the limiting factor in the operation of these systems. In a heat engine the most common condensed corrodent is Na2SO4 whereas in a heat exchanger an oxide slag may be present. The primary emphasis is on Na2SO4 induced corrosion, however, similarities and differences to oxide slag are also discussed. The extensive research on corrosion of metal alloys has led to understanding and controlling corrosion for these materials. Currently silicon based ceramics are prime candidates for the applications discussed. Therefore it is important to understand the effects of condensed phase deposits on this emerging class of high temperature materials. Both the thermodynamic and strength of the ceramic is also examined. Finally some control strategies for corrosion of silicon based ceramics are explored