Let [Formula: see text] be a finite geometric separable extension of the rational function field [Formula: see text], and let [Formula: see text] be a finite cyclic extension of [Formula: see text] of prime degree [Formula: see text]. Assume that the ideal class number of the integral closure [Formula: see text] of [Formula: see text] in [Formula: see text] is not divisible by [Formula: see text]. Using genus theory and Conner–Hurrelbrink exact hexagon for function fields, we study in this paper the [Formula: see text]-class group of [Formula: see text] (i.e. the Sylow [Formula: see text]-subgroup of the ideal class group of [Formula: see text]) as Galois module, where [Formula: see text] is the integral closure of [Formula: see text] in [Formula: see text]. The resulting conclusion is used to discuss the relations of class numbers for the biquadratic function fields with their quadratic subfields.
Traditional image interpolation methods always are introduced by piecewise-cubic kernel, which have been derived in 1D with one parameter and applied to 2D images in a separable way. However, images typically are statistically nonseparable, and it motivates the construction of nonseparable cubic kernel function. This paper derives a new nonseparable cubic kernel based on the Coons-type surface method, then the efficiency of our method is compared with that of the traditional algorithm of cubic kernel for image resizing in Matlab 7. The experimental results show that our proposed algorithm excels the bicubic interpolation method in visual effect and complexity.
In this paper, we show that the Gilbert-Varshamov and the Xing bounds can be improved significantly around two points where these two bounds intersect by nonlinear codes from algebraic curves over finite fields.
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