Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pyth...
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Diophantine equations have always fascinated mathematicians about existence, finitude, and the calculation of possible solutions. Among these equations, one of them will be the object of our research. This is the Pythagoras’- Fermat’s equation defined as follows. (1) when , it is well known that this equation has an infinity of solutions but has none (non-trivial) when . We also know that the last result, named Fermat-Wiles theorem (or FLT) was obtained at great expense and its understanding remains out of reach even for a good fringe of professional mathematicians. The aim of this research is to set up new simple but effective tools in the treatment of Diophantine equations and that of Pythagoras-Fermat. The tools put forward in this research are the properties of the quotients and the Diophantine remainders which we define as follows. Let a non-trivial triplet () solution of Equation (1) such that . and are called the Diophantine quotients and remainders of solution . We compute the remainder and the quotient of b and c by a using the division algorithm. Hence, we have: and et with . We prove the following important results. if and only if and if and only if . Also, we deduce that or for any hypothetical solution . We illustrate these results by effectively computing the Diophantine quotients and remainders in the case of Pythagorean triplets using a Python program. In the end, we apply the previous properties to directly prove a partial result of FLT. .
In order to solve the parallel algorithm for Petri nets system with concurrent function, to realize the parallel control and execution of the Petri nets, the Petri nets parallel subnets conditions were proposed, that ...
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In order to solve the parallel algorithm for Petri nets system with concurrent function, to realize the parallel control and execution of the Petri nets, the Petri nets parallel subnets conditions were proposed, that provides the theory basis for judging P- invariant whether was the parallel subnet. Firstly, we according to the concurrent character of Petri net model, to analyze the parallelism of Petri net system, Pinvariants the solving process and it's subnet division were given;Secondly, the Petri nets parallel subnets conditions were proposed, gives P- invariant constituted of Petri net parallel subnets decision theorem, and the theoretical proof and example verification;Finally, A Petri net parallel subnet division algorithm based on P- invariants were given. Theoretical validate and experimental result shows that Petri nets parallel subnets conditions set and divided algorithm were correct and effective.
This paper focuses on a peak force milling control system that is an integral component of a real-time process monitoring and control system for CNC milling. In process planning, reference peak forces are selected for...
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ISBN:
(纸本)9781509034680
This paper focuses on a peak force milling control system that is an integral component of a real-time process monitoring and control system for CNC milling. In process planning, reference peak forces are selected for each tool move for safe, accurate, and efficient machining. Peak forces are selected based on constraints such as dimensional tolerance and desired chip load. The peak force controller adjusts the feedrate during machining to maintain the reference peak force. This is very important as the tool wears or as unexpected workpiece geometry is encountered. The peak force controller is based on a nonlinear division algorithm that adjusts the feedrate based on a ratio of the desired force to the measured peak force. The stability and performance of this algorithm is investigated by analysis, simulation, and experiment. Describing function analysis of the linearized system shows stable controller behavior for excessive tool wear and unexpected cutting geometry. Simulations using the complete nonlinear equations, along with experiments, verify the good performance of the controller for a number of workpiece materials, including aluminum, steel and Inconel.
The K-NN query algorithm is an important class of query algorithm in spatial database, the traditional K-NN query algorithm used the measurement distance and pruning strategy to search in the adopted index tree, the r...
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ISBN:
(纸本)9781424480333
The K-NN query algorithm is an important class of query algorithm in spatial database, the traditional K-NN query algorithm used the measurement distance and pruning strategy to search in the adopted index tree, the regional division algorithm based on the grid obtained the nearest neighbor objects by calculating the distance between the queried object and the data objects within the eight grids around the queried object, and carried on expanding inquiry scope outward layer by layer, in view of the advantage of the grid index, a new K-NN query algorithm based on the dynamic movement of spatial objects is proposed in this paper, it only carries on grid division to a small part of region around the queried object, when all the data objects in the current grid are already searched, make the cells in the grid empty, dynamically move the external data objects into the grid area continue to query according to the relative position of these objects, because the algorithm uses a fixed grid area to query the data objects in the whole region, so it can reduce the target storage space and the complexity degree of the algorithm calculation, the experiments show that the new query algorithm is superior to the traditional algorithms in the querying performance when the amount of data is increased, and the query efficiency is greatly increased.
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