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对交互式质询来说，因数分解足以胜任。Factorization works well enough for interactive challenges.

设计一个因子分解算法，并分析其复杂性。Design a factorization algorithm and analysis of their complexity.

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用试凑法对这个方程进行因式分解是不可能奏效的。Factorization of this equation by trial and error is not possible.

注意，同一个素数在因数分解中可以出现多次。Note that, in the same prime factorization can appear multiple times.

RSA算法的安全性依赖于大数的因数分解的困难性。The safe of RSA algorithm based on difficulty in the large number factorization.

这类问题与矩阵函数因子分解有着密切联系。The problems are closelurelated with factorization problems for rational matrix functions.

该算法利用QR因式分解以解决最小二乘方探测问题。The algorithm relies on a QR factorization approach to the least-squares detection problem.

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第二部分研究了代数与余代数之间的缠扭结构以及与其密切相关的代数分解理论。In Part Two we study the theory of entwining structures and that of factorization structures.

本文提出了基于复数U-D分解的复参数最小二乘估计方法。The Paper presents a new U-D factorization based least squares methods for complex estimation.

提出了基于复数U-D分解的复参数最小二乘估计方法。The paper presents a new U-D factorization based least squares methods for complex estimation.

第五章，研究了高阶奇异系统的实现与因子分解问题。In chapter 5, the realization and factorization of higher-order singular systems have been studied.

公钥密码算法RSA主要是依赖于大数的因数分解的困难性建立的。Public key RSA mainly depends on the establishment of the factorization difficulty of a big integer.

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目前在公开金钥加密法中，最常用到的RSA密码演算法，就是应用因数分解的原理。The popular RSA cipher algorithm, widely deployed in public-key cryptography, relies on factorization.

一旦因子分解问题可解，攻击者就可以伪造签名。An attacker could generate a forged signature on the assumption when the factorization problem is solved.

更进一步地，我们将LU分解表示成1-带状矩阵的乘积并得到它的反矩阵。Furthermore, we express the LU factorization into 1-banded factorizations and get the inverse explicitly.

这种技术之所以安全，是因为应用了因数分解或其他困难的数学问题。The security of public-key cryptography depends on factorization or other difficult mathematical problems.

在隐式NND格式中引入了一种分解算法，提高了计算效率。Moreover, a factorization algorithm is introduced to implicit NND scheme to raise computational efficiency.

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偏微分方程使用分离变量法和对称化简法求解。Factorization methods for higher-order differential equations use techniques due to Bronstein and van Hoeij.

对差分方程之隐式部分利用了近似因式分解法。Jacobian Matrix splitting and approximate factorization are used for solving obtained implicit difference equations.

通过综合应用反推、调节函数、SDU分解和估计参数，得到了控制器的设计。The control design is achieved by using backstepping, tuning functions, SDU factorization and estimating parameters.