• Journal of the Chungcheong Mathematical Society
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• v.26 no.4
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• pp.853-867
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• 2013

In this paper we study the cardinality of the dot product set generated by two subsets of vector spaces over finite fields. We notice that the results on the dot product problems for one set can be simply extended to two sets. Let E and F be subsets of the d-dimensional vector space $\mathbb{F}^d_q$ over a finite field $\mathbb{F}_q$ with q elements. As a new result, we prove that if E and F are subsets of the paraboloid and ${\mid}E{\parallel}F{\mid}{\geq}Cq^d$ for some large C > 1, then ${\mid}{\Pi}(E,F){\mid}{\geq}cq$ for some 0 < c < 1. In particular, we find a connection between the size of the dot product set and the number of lines through both the origin and a nonzero point in the given set E. As an application of this observation, we obtain more sharpened results on the generalized dot product set problems. The discrete Fourier analysis and geometrical observation play a crucial role in proving our results.

• The Transactions of the Korea Information Processing Society
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• v.7 no.10
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• pp.3098-3104
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• 2000

This paper defiens a perceptron-like self-organizing map(PSOM) and show that PSOM is equivalent to Kohonen's self-organizing map(SOM) if target values of output neurons of PSOM are selected properly. This fact imphes that PSOM is a generalized SOM algorithm. This paper also show that if clustering is restricted to vector sets distributed on hypersphere with unit radius, SOM and dot-product SOM(DOSM) are equivalent algorithms. Therefore we conclude that DSOM is a special case of SOM, which in turn a special, case of PSOM.