Quasi-Chun-Ching Shih''''s Fractional Fourier Transform with Periodicity of 2,3 and M

Based on Chun-Ching Shih's idea, the basic transform was substituted and the quasi-ChunChing Shih's fractional Fourier transform with periodicity of 2, 3 and M was deduced. The two former transforms and the Chun-Ching Shih's fractional Fourier transform were only the particular cases of quasiChun-Ching Shih's fractional Fourier transform with periodicity of M.     

Covering a string

We consider the problem of finding the repetitive structures of a given stringx. The periodu of the stringx grasps the repetitiveness ofx, sincex is a prefix of a string constructed by concatenations ofu. We generalize the concept of repetitiveness as follows: A stringw covers a stringx if there is a superstring ofx which is constructed by concatenations and superpositions ofw. A substringw ofx is called aseed ofx ifw coversx. We present anO(n logn)-time algorithm for finding all the seeds of a given string of lengthn.Partially supported by SERC Grants GR/F 00898 and GR/J 17844, NATO Grant CRG 900293, ESPRIT BRA Grant 7131 for ALCOMII, and MRC Grant G 9115730.Partially supported by MRC Grant G 9115730 and S.N.U. Posco Research Fund 94-15-1112.     

Transition of the flow in a symmetric channel with periodically expanded grooves

Transition of the flow in a periodically grooved channel is numerically investigated for periodicity indices m=1 up to 6 by assuming the two-dimensional and fully developed flow field, where m is defined as a number of grooves in which the flow repeats periodically. Critical Reynolds numbers for the onset of a self-sustained oscillatory flow from a steady-state flow are evaluated by numerical simulations. It is found that the bifurcations occur at the critical Reynolds numbers as a result of Hopf bifurcation, and a period in the streamwise direction of the oscillatory flow is twice as long as the groove pitch of the channel. In addition, flow visualization with the aluminum dust method is carried out to confirm the results obtained from the numerical simulations. The experimental results are in good agreement with the numerical ones.     

Computing maximal-exponent factors in an overlap-free word

The exponent of a word is the quotient of its length over its smallest period. The exponent and the period of a word can be computed in time proportional to the word length. We design an algorithm to compute the maximal exponent of all factors of an overlap-free word. Our algorithm runs in linear-time on a fixed-size alphabet, while a naive solution of the question would run in cubic time. The solution for non-overlap-free words derives from algorithms to compute all maximal repetitions, also called runs, occurring in the word.We also show there is a linear number of occurrences of maximal-exponent factors in an overlap-free word. Their maximal number lies between 0.66n and 2.25n in a word of length n. The algorithm can additionally locate all of them in linear time.     

Discretization behavior analysis of a switcbing control system from a unified mathematical approach

A useful unified analysis framework is proposed for exploring the intriguing behaviors of a second-order switching control system. Complex discretizafion behaviors of the switching control system are explored in detail, and some intrinsic relationships between the system periodic behaviors and their associated symbolic sequences are studied.     

CHARACTERISTICS OF MICROCUTTING FORCE VARIATION IN ULTRAPRECISION DIAMOND TURNING

An investigation of the characteristics of microcutting forces in diamond turning of crystalline materials is presented. The characteristics of the cutting forces were extracted and analyzed using statistical and spectrum analysis methods. A series of cutting experiments were done on a copper alloy and copper single crystals with different crystallographic orientations. Experimental results indicate that there exists a dominant frequency component and a periodicity of fluctuation of the cutting forces per workpiece revolution in the diamond turning of a single crystal material. The periodicity is closely related to the crystallographic orientation of the material being cut. As the depth of cut increases, the influence of crystallographic orientation of the single-crystal materials on microcutting forces is found to be more pronounced. Moreover, the cutting force ratio between the mean thrust force and the mean cutting force is found to vary with the depth of cut, and a large ratio was observed at a small depth of cut. These findings help to explain quantitatively the periodic fluctuations of microcutting forces (and hence the materials-induced vibration) in ultraprecision diamond turning, which are not encountered in conventional machining.     

Binary Sequences Derived from Dickson Permutation Polynomials over Binary Extension Field

In this paper, based on a result of Lidl and Mullen (Mathematical Journal of Okayama University, 1991), the maximum length and the second maximum length that can be attained by cycles of Dickson permutation polynomial (of the first kind) with parameter 1 are studied. Necessary and sufficient conditions for these two lengths to be attained are given, which are connected with Fermat primes and Mersenne primes, respectively. Furthermore, a class of coordinate sequences that maintains a large period is obtained, which is shown to be the coordinate sequences derived from cycles of the second maximum length. Explicit formulas for their periodicity and shift-equivalences are also presented.