R70-43 Lattice Functions, Pair Algebras, and Finite-State Machines
This paper is yet another solution for finding linear internal state assignments for the next-state behavior of a sequential machine. A state assignment is said to be linear if and only if the resulting next-state variables are linear combinations ofthe present-state and input variables. Consequently, the next-state behavior of a sequential machine is said to be linear if and only if there exists a one-to-one assignment which is linear. The problem of finding linear assignments has received considerable attention, probably due to their inherent mathematical structure, and hopefully to further illuminate the theory of sequential machines in general. The prime contribution of this paper is a refinement of a previously presented algorithm [1] for finding linear assignments. In this algorithm an assignment is found for a single column, as an autonomous machine, and it is then checked for validity with the remaining columns. The problem, of course, is to be able to restrict the choice of possible assignments so that the number that has to be checked for validity is sufficiently small to keep the process from becoming an exhaustive search. This the author has managed to do, with reasonable success. The author does go into considerable detail in explaining the background and individual steps of the algorithm. While all of the detail may be useful, it does make the paper significantly longer than it needs to be. The prime criticism of the paper is not in the algorithm itself, but in what the author purports that it will do. In the first instance, it is stated that the algorithm is very effective even when the number of states is large. If the matrix A is not unique, then it is necessary to try each possible A in the subsequent steps of the algorithm. In such circumstances, an algorithm can hardly be termed effective. Furthermore, the method requires the graph of the matrix A, which is no easy task for large n (the order of A). As an example, if the cycle length is 19, then the order ofA is 18 and the graph contains 218 = 262 144 vectors of length 18, and yet a next-state table with 19 states is not at all unreasonable. It is therefore concluded that while the algorithm will work, it may become exceedingly tedious for some cases. In the second instance, the author restricts his algorithm to a realization oforder n, where the number of states is q, and 2n' < q< 2'. As an example, the following machine (machine l) has ten states and hence, following the author, n= 4. In contradiction, however, when the algorithm is applied, n =6. Obviously the restriction is fallacious and it is only necessary for q< 2'. The author has omitted a few rather important references [2]-[4], not all of which were available at the time of writing; however, they are mentioned for completeness.
Linearity of Sequential Machines
A sequential machine is defined to be linear if its next-state function and output function are linear transformations from their domains (vector spaces) to their ranges (also vector spaces). It is shown that this definition is equivalent to those given by other authors. Based on this definition, a set of necessary and sufficient conditions for the flow table of a sequential machine to be linear is obtained. Many properties of the flow table of a linear sequential machine are found, and in many cases they form very simple tests for the linearity of a flow table. A general procedure for testing the linearity of a flow table is established. This procedure, including the coding of the states, inputs, and outputs, either ends with a linear realization of the flow table with the minimum possible numbers of state variables, input variables, and output variables, or detects that such a linear realization for the flow table is impossible. The type of sequential machine considered in this paper is deterministic and synchronous, and both Moore model and Mealy model are studied in detail. The linearity of incompletely specified sequential machines is also discussed.
Analysis of the stress wave effect during coal breakage by a high-pressure abrasive air jet
In view of the defects of borehole collapse, inhibition of gas desorption and migration of gas existing in hydraulic fracturing and other hydraulic permeability–increasing measures for soft coal seams with low-permeability technology is proposed for coal breakage by a high-pressure abrasive gas jet for relieving pressure and increasing permeability. The comparative analysis of gas jet flow field structure between convergent nozzle and Laval nozzle has been given by numerical simulation. For Laval nozzle, the expansion wave and compression wave alternate and move forward steadily in gas jet and vanish when potential core length reaches maximum. So, the Laval nozzle can form more stable flow filed structure of gas jet and avoid shock wave in gas jet. Furthermore, a high-speed camera is adopted to analyze the jet structure and verify the conclusion of numerical simulation. Based on thermodynamic theory, this article calculates and analyzes the critical local sound velocity and pressure generated from the stress wave during the process of coal breakage by the gas jet. Furthermore, experimental coal breakage by a high-pressure abrasive gas jet is carried out. The high-pressure abrasive gas jet impacts the coal body as a quasi-static load and a dynamic load and forms corrosion pits on the surface of the coal body. Penetrating cracks are formed within the coal in the pattern of the loaded stress wave which leads to coal breakage. The effects of porosity and permeability on the propagation of the stress wave in coal are analyzed by establishing the dispersion equation for the spread of the stress wave in coal. The results show that porosity has a significant effect on wave velocity and that the attenuation of the stress wave is intensified with an increase in porosity. Moreover, the stress wave attenuation is more obvious at high frequency. The effect of permeability on the wave velocity is not significant at low frequencies. In contrast, at high frequency and relatively low permeability, the wave velocity increases with the permeability, and the attenuation of the wave velocity initially increases and then decreases. When the permeability is greater than 10−11 m2, the wave velocity is not affected by the permeability. However, the stress wave is not attenuated.