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Let $ N $ be a sufficiently large integer. In this paper, it is proved that, with at most $ O\big(N^{4/9+\varepsilon}\big) $ exceptions, all even positive integers up to $ N $ can be represented in the form $ p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^3 $, where $ p_1, p_2, p_3, p_4, p_5, p_6 $ are prime numbers.

Let N be a sufficiently large real number. In this paper, it is proved that, for $$1

Let N be a sufficiently large real number. In this paper, we prove that, for $$10$$ , the Diophantine inequality $$\begin{aligned} \big |p_1^c+p_2^c+p_3^c-N\big

Let $$\mathcal {P}_r$$ denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, we establish a theorem of Bombieri–Vinogradov type for the Piatetski–Shapiro primes $$p=[n^{1/\gamma }]$$ with $$\frac{85}{86}

Let N be a sufficiently large integer. In this paper, it is proved that, with at most O(N 119/270+s) exceptions, all even positive integers up to N can be represented in the form where p1, p2, p3, p4, p5, p6 are prime numbers.

Let $$\mathcal {P}_r$$ denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, we generalize the result of Vaughan (Bull Lond Math Soc 17(1):17–20, 1985) for ternary ‘admissible exponent’. Moreover, we use the refined ‘admissible exponent’ to prove that, for $$3\leqslant k\leqslant 14$$ and for every sufficiently large even integer n, the following equation $$\begin{aligned} n=x^2+p_1^2+p_2^3+p_3^3+p_4^3+p_5^k \end{aligned}$$ is solvable with x being an almost-prime $$\mathcal {P}_{r(k)}$$ and the other variables primes, where r(k) is defined in Theorem 1.1. This result constitutes a deepening of previous results.

Let N be a sufficiently large integer. We prove that almost all sufficiently large even integers n ∈ [N − 6U, N + 6U] can be represented as $$\left\{ {\matrix{ {n = p_1^2 + p_2^3 + p_3^3 + p_4^3 + p_5^3 + p_6^3} \hfill \cr {\left| {p_1^2 - {N \over 6}} \right| \le U,\,\,\,\left| {p_i^3 - {N \over 6}} \right| \le U,\,\,\,i = 2,3,4,5,6} \hfill \cr } } \right.$$ where U = N1−δ+e with δ ⩽ 8/225.

A new method is presented to determine parameter values (knot) for data points for curve and surface generation. With four adjacent data points, a quadratic polynomial curve can be determined uniquely if the four points form a convex polygon. When the four data points do not form a convex polygon, a cubic polynomial curve with one degree of freedom is used to interpolate the four points, so that the interpolant has better shape, approximating the polygon formed by the four data points. The degree of freedom is determined by minimizing the cubic coefficient of the cubic polynomial curve. The advantages of the new method are, firstly, the knots computed have quadratic polynomial precision, i.e., if the data points are sampled from a quadratic polynomial curve, and the knots are used to construct a quadratic polynomial, it reproduces the original quadratic curve. Secondly, the new method is affine invariant, which is significant, as most parameterization methods do not have this property. Thirdly, it computes knots using a local method. Experiments show that curves constructed using knots computed by the new method have better interpolation precision than for existing methods.

Let N be a sufficiently large integer. In this paper, it is proved that, with at most $$O(N^{7/18+\varepsilon })$$ exceptions, all even positive integers up to N can be represented in the form $$p_1^2+p_2^2+p_3^3+p_4^3+p_5^4+p_6^4$$ , where $$p_1,p_2,p_3,p_4,p_5,p_6$$ are prime numbers, which constitutes an improvement over some previous work.

Let N be a sufficiently large integer. In this paper, it is proved that, with at most $$O(N^{4/27+\varepsilon })$$ exceptions, all even positive integers up to N can be represented in the form $$p_1^2+p_2^2+p_3^3+p_4^3+p_5^6+p_6^6$$ , where $$p_1,p_2,p_3,p_4,p_5,p_6$$ are prime numbers. This gives a large improvement of a recent result $$O(N^{127/288+\varepsilon })$$ due to Liu (Proc. Indian Acad. Sci. (Math. Sci.) 130(1) (2020) Article ID. 8).

Let N be a sufficiently large real number. In this paper, it is proved that for 1 c 6 / 5 , the Diophantine inequality | p 1 c + p 2 c + p 3 c + p 4 c − N | log − 1 ⁡ N is solvable in prime numbers p 1 , p 2 , p 3 , p 4 . This result constitutes an improvement upon that of Mu (2015) [15] for the range 1 c 97 / 81 .

This paper proposes a contour extraction model based on cosaliency detection for remote sensing image airport detection and improves the traditional line segmentation detection (LSD) algorithm to make it more suitable for the goal of this paper. Our model consists of two parts, a cosaliency detection module and a contour extraction module. In the first part, the cosaliency detection module mainly uses the network framework of Visual Geometry Group-19 (VGG-19) to obtain the result maps of the interimage comparison and the intraimage consistency, and then the two result maps are multiplied pixel by pixel to obtain the cosaliency mask. In the second part, the contour extraction module uses superpixel segmentation and parallel line segment detection (PLSD) to refine the airport contour and runway information to obtain the preprocessed result map, and then we merge the result of cosaliency detection with the preprocessed result to obtain the final airport contour. We compared the model proposed in this article with four commonly used methods. The experimental results show that the accuracy of the model is 15% higher than that of the target detection result based on the saliency model, and the accuracy of the active contour model based on the saliency analysis is improved by 1%. This shows that the model proposed in this paper can extract a contour that closely matches the actual target.

The main purpose of this paper is to use the Hardy&ndash, Littlewood method to study the solvability of mixed powers of primes. To be specific, we consider the even integers represented as the sum of one prime, one square of prime, one cube of prime, and one biquadrate of prime. However, this representation can not be realized for all even integers. In this paper, we establish the exceptional set of this kind of representation and give an upper bound estimate.

Let d(n) denote the Dirichlet divisor function. Define $$\begin{aligned} \mathcal {S}_k(x,y):= \sum _{\begin{array}{c} x-y

Suppose that λ 1 and λ 2 are positive real numbers such that λ 1 / λ 2 is irrational and algebraic. Let V be a well-spaced sequence and δ > 0 . Denote by E ( V , X , δ ) the number of v ∈ V with v ≤ X such that the inequality | λ 1 p 1 + λ 2 p 2 − v | v − δ has no solution in primes p 1 , p 2 . We prove that for all X ≥ 1 , E ( V , X , δ ) ≪ X f ( δ ) + e for any e > 0 with f ( δ ) = max ⁡ ( 5 / 9 + 2 δ , 2 / 3 + 4 δ / 3 ) , which improves the earlier result.

Let $1, Comment: 21 pages

Let [Formula: see text] denote an almost-prime with at most [Formula: see text] prime factors, counted according to multiplicity. In this paper, it is proved that for every sufficiently large even integer [Formula: see text], the equation [Formula: see text] is solvable with [Formula: see text] being an almost-prime [Formula: see text] and the other variables primes. This result constitutes an improvement upon that of Cai, who obtained the same conclusion, but with [Formula: see text] in place of [Formula: see text].

Let N be a sufficiently large integer. In this paper, we prove that almost all sufficiently large even integer n ∈ [N − 6U,N + 6U] can be represented as n = p12 + p 22 + p 33 + p 43 + p 54 + p 64, pi2 −N 6 ≤ U,i = 1, 2, pi3 −N 6 ≤ U,i = 3, 4, pi4 −N 6 ≤ U,i = 5, 6, where U = N1−δ+𝜀 with δ = 1/44.

Let [Formula: see text] denote an almost-prime with at most [Formula: see text] prime factors, counted according to multiplicity. In this paper, it is proved that, for [Formula: see text] and for every sufficiently large even integer [Formula: see text], the equation [Formula: see text] is solvable with [Formula: see text] being an almost-prime [Formula: see text] and the other variables primes, which constitutes an extension upon that of Lü.

In this paper, we study the hybrid problem of Hua’s theorem and the Piatetski-Shapiro prime number theorem, and obtain results in this direction of the nonhomogeneous case [Formula: see text], which deepen the classical result of Hua.

We prove that there are infinite cube-free numbers of the form [n c ] for any fixed real number c ∈ (1, 11/6).

Let [Formula: see text] be the error term of the Dirichlet divisor problem. In this paper, we establish an asymptotic formula of the seventh-power moment of [Formula: see text] and prove that [Formula: see text] with [Formula: see text] which improves the previous result.

Let $N$ be a sufficiently large integer. In this paper, it is proved that with at most $O(N^{13/16+\varepsilon})$ exceptions, all even positive integers up to $N$ can be represented in the form $p_1^2 + p_2^2 + p_3^3 + p_4^3 + p_5^4 + p_6^4$, where $p_1$, $p_2$, $p_3$, $p_4$, $p_5$, $p_6$ are prime numbers.

Let $N$ be a sufficiently large real number. In this paper, it is proved that, for $1, Comment: 13 pages. arXiv admin note: substantial text overlap with arXiv:1810.09368

In this paper, we propose a new robust feature extraction algorithm for 3D models based on principal curvature direction. Generally, the feature regions tend to be more noisy, so it demands a robust technique to handle features effectively. Because the integral invariants are robust against noise, the principal curvature information is estimated based on principal component analysis. After fuzzy filtering of the principal curvature direction, it becomes a good description of the geometric discontinuity. Compared with the curvature values, the impact of noise on the principal curvature direction is small. Therefore, feature extraction based on principal curvature direction is more robust and accurate. The experimental results show that the proposed algorithm can efficiently extract feature and distinguish noise.

Let $d(n;\ell_1,M_1,\ell_2,M_2)$ denote the number of factorizations $n=n_1n_2$, where each of the factors $n_i\in\mathbb{N}$ belongs to a prescribed congruence class $\ell_i\bmod M_i\,(i=1,2)$. Let $\Delta(x;\ell_1,M_1,\ell_2,M_2)$ be the error term of the asymptotic formula of $\sum\limits_{n\leqslant x}d(n;\ell_1,M_1,\ell_2,M_2)$. In this paper, we establish an asymptotic formula of the fourth power moment of $\Delta(M_1M_2x;\ell_1,M_1,\ell_2,M_2)$ and prove that \begin{equation*} \int_1^T\Delta^4(M_1M_2x;\ell_1,M_1,\ell_2,M_2)\mathrm{d}x=\frac{1}{32\pi^4}C_4\Big(\frac{\ell_1}{M_1},\frac{\ell_2}{M_2}\Big) T^2+O(T^{2-\vartheta_4+\varepsilon}), \end{equation*} with $\vartheta_4=1/8$, which improves the previous value $\theta_4=3/28$ of K. Liu., Comment: 21 pages

This paper constructs a dyadic non-subsampled Contourlet transform for denoising on the image, the transformation has more directional subband, using the non-subsampled filter group for decompositing of direction, so has the translation invariance, eliminated image distortion from Contourlet transform’s lack of translation invariance. Non-subsampled filter reduces noise interference and data redundancy. Using the feature of NSCT translation invariance, multiresolution, multi-direction, and can according to the energy of NSCT in all directions and in all scale, adaptive denoising threshold. Experimental results show that compared to wavelet denoising and traditional Contourlet denoising, the method achieves a higher PSNR value, while preserving image edge details, can effectively reduce the Gibbs distortion, improve visual images.

Existing image inpainting algorithm based on low-rank matrix approximation cannot be suitable for complex, large-scale, damaged texture image. An inpainting algorithm based on low-rank approximation and texture direction is proposed in the paper. At first, we decompose the image using low-rank approximation method. Then the area to be repaired is interpolated by level set algorithm, and we can reconstruct a new image by the boundary values of level set. In order to obtain a better restoration effect, we make iteration for low-rank decomposition and level set interpolation. Taking into account the impact of texture direction, we segment the texture and make low-rank decomposition at texture direction. Experimental results show that the new algorithm is suitable for texture recovery and maintaining the overall consistency of the structure, which can be used to repair large-scale damaged image.

The essence of image segmentation is a based on some properties the process for pixel classification. Firstly, three typical methods in image segmentation methods are outlined and their characteristic is analyzed in this paper. The traditional visual attention model is described and improved in this paper. The input image gray value and edge features are extracted by Gabor filters and the Gauss - Laplace operator, gray feature maps and the edge feature maps are got respectively, then interest regions image is obtained by linear combination. The interest region in interest region image is selected by the dynamic neural network methods in artificial intelligence. The limit scope of regional growth is provided improved visual attention model algorithm identified interest region, the binary image is got by setting gray-value. At last, image segmentation is achieved by image segmentation algorithm based improved visual attention model and region growing. Experimental results validate that this methods not only achieve image segmentation, but also accurately and automatically achieve interest region segmentation, improve the quality of the segmentation, and has good robustness.