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Let 𝔽 q {\mathbb{F}_{q}} be the finite field of q = p m ≡ 1 ( mod 4 ) {q=p^{m}\equiv 1~{}(\bmod~{}4)} elements with p being an odd prime and m being a positive integer. For c , y ∈ 𝔽 q {c,y\in\mathbb{F}_{q}} with y ∈ 𝔽 q * {y\in\mathbb{F}_{q}^{*}} non-quartic, let N n ⁢ ( c ) {N_{n}(c)} and M n ⁢ ( y ) {M_{n}(y)} be the numbers of zeros of x 1 4 + ⋯ + x n 4 = c {x_{1}^{4}+\cdots+x_{n}^{4}=c} and x 1 4 + ⋯ + x n - 1 4 + y ⁢ x n 4 = 0 {x_{1}^{4}+\cdots+x_{n-1}^{4}+yx_{n}^{4}=0} , respectively. In 1979, Myerson used Gauss sums and exponential sums to show that the generating function ∑ n = 1 ∞ N n ⁢ ( 0 ) ⁢ x n {\sum_{n=1}^{\infty}N_{n}(0)x^{n}} is a rational function in x and presented its explicit expression. In this paper, we make use of the cyclotomic theory and exponential sums to show that the generating functions ∑ n = 1 ∞ N n ⁢ ( c ) ⁢ x n   and   ∑ n = 1 ∞ M n + 1 ⁢ ( y ) ⁢ x n \sum_{n=1}^{\infty}N_{n}(c)x^{n}\quad\text{and}\quad\sum_{n=1}^{\infty}M_{n+1}% (y)x^{n} are rational functions in x. We also obtain the explicit expressions of these generating functions. Our result extends Myerson’s theorem gotten in 1979.

Let Fq be the finite field of q = p m ≡ 1 (mod 4) elements with p being an odd prime and m being a positive integer. For c, y ∈ Fq with y ∈ F * q nonquartic, let Nn(c) and Mn(y) be the numbers of zeros of x 4 1 + ... + x 4 n = c and x 4 1 + ... + x 4 n−1 + yx 4 n = 0, respectively. In 1979, Myerson used Gauss sum and exponential sum to show that the generating function ∞ n=1 Nn(0)x n is a rational function in x and presented its explicit expression. In this paper, we make use of the cyclotomic theory and exponential sums to show that the generating functions ∞ n=1 Nn(c)x n and ∞ n=1 M n+1 (y)x n are rational functions in x. We also obtain the explicit expressions of these generating functions. Our result extends Myerson's theorem gotten in 1979.

Recently, deep learning has introduced new prospects to numerous practical applications such as image recognition, robot navigation, gene engineering, language processing, and traffic sign identification. Several network models including AlexNet, VGGNet, GoogLenet and ResNet, have achieved milestone contributions while relying on massive computing resources. However, when faced with a small number of labeled examples, especially in the case of unbalanced datasets, the cumulative error and time-consuming convergence reduce their efficacy. Inspired by the convolutional output layer with a [Formula: see text] kernel, a convolutional nonlinear transfer approach with partial cooperating (CNN-COL) is proposed to address this challenge. Meanwhile, a novel method for data augmented balance can enhance the influence of small and unbalanced samples in the CNN-COL. Related experiments show that the proposed CNN-COL can effectively improve the quality of a dataset and achieve superior performance with respect to traffic sign identification based on a small and type-unbalanced dataset.