对偶的

This gives a new description of human recognition mechanism from the view of information geometrical and algebraic topological treatment, the results include Propose a structure representation and system identification approach for neural networks, based on the concept of simplical complexes and dual affine coordinates in the dual flat manifold.

主要的结果包括：提出用对偶平坦流形上的单、复形结构和对偶平坦流形的对偶仿射坐标作为神经网络内在结构表示的一种方法，具有广泛的意义。

Next, we discuss the relations between left quasi-dual bimodules and left dual-bimodules, we obtain that a left quasi-dual bimodule is a left dual bimodule if it satisfies one of the following conditions: sM is minimal injective and MR is a M-minimal injective kasch-module; MR is a M-minimal injective kasch-module and for any two ideals LI and L2 ofSS rM(L1 n L2)-rw(L1)+rM(I2); sM is minimal injective and for any two submodules A and B of MR,Lastly, we applicate the quasi-duality on smash product algebra R#H, and obtain an answer of the semiprime problem, i.e., let H be a finite-dimensional semisimple Hopf algebra and R be an H-module algebra, if R is left quasi-dual and semiprime, then R#H is semiprime.

我们得到：一个左拟对偶双边模如果满足下列条件之一，则它将成为一个左对偶双边模：_sM是单内射的并且M_R是一个M-单内射kasch-模；M_R是一个M-单内射kasch-模并且对_sS的任意两个理想，有r_M(L_1∩L_2)=r_M(L_1)+r_M(L_2)；_sM是单内射的且对M_R的任意两个子模，有l_s=l_s+l_s。2 在第2.3节中我们将拟对偶性应用于smash积代数R#H，部分解决了半素问题。

In this paper,the author introduced an approximate augmented Lagrangian function in nonlinear programming,established dual mapping and the related dual problem of this augmented Lagrangian function and obtained the results of approximate strong duality and approximate weak duality of original problems.

介绍了非线性规划中的一种近似增广拉格朗日函数，建立了基于这种增广拉格朗日函数的对偶映射和相应的对偶问题，得到了原问题和对偶问题的强近似对偶和弱近似对偶结果。

Following the dual principle of the circuit theories,the dual network of mutual inductance network ,the concept of mutual capacitance,and the dual relation of mutual capacitance and mutual indutance are analysed and substantiated,and the decoupling equivalent circuit of mutual capacitance is discussed as well.

从电路理论对偶原理出发，分析论证了互感网络的对偶网路、互电容的概念及互电容与互电感的对偶关系，并探讨了互电容的去耦等效电路。

At length 50, we got 3 inequivalent extremal self-dual [50,25,10] codes.

将这个性质用在寻找长度为66的自对偶码，我们得到至少五十个对应於w2互不等价的极端自对偶[66，33，12]码与一个对应於w1，β=32的极端自对偶[66，33，12]码；用在寻找长度为50的自对偶码上，我们得到三个互不等价的极端自对偶[50，25，10]码。

Furthermor, we consider their nondiferentiable situation, we define nonsmooth univex functions for Lipschitz functions by using Clarke generalized directional derivative and study nonsmooth multiobjective fractional programming with the new convexity. We establish generalized Karush-Kuhn-Tucker necessary and sufficient optimality condition and prove weak, strong and strict converse duality theorems for nonsmooth multiobjective fractional programming problems containing univex functions.

而且，本文利用Clarke广义方向导数针对Lipschitz函数在原来一致凸函数概念的基础上定义了不可微的一致凸函数，并利用这类新凸性，我们研究了非光滑多目标分式规划，获得了广义Karush-Kuhn-Tucher最优性条件；弱对偶定理、强对偶定理和严格逆对偶定理。

The strong and weak dual theorems are obtained in the sense of strictly efficiency in dual model which is established by scalar set-valued Lagrange mapping.

利用标量集值Lagrange映射建立了集值优化问题的对偶模型，并得到严有效性下的弱对偶和强对偶定理。

In the sense of weak effective solution,the problems of Lagrange duality and saddle point are discussed under the convexlike condition.An alternative theorem of the system containing a semidefinite constraint of a matrix function is established at first.

首先在似凸条件下建立了一个含矩阵函数半定约束系统的择一性定理，由此得到多目标半定规划及其在弱有效解意义下的Lagrange对偶理论，包括弱对偶、强对偶和逆对偶等。

In section 3, on the base of the iterative algorithm, we obtain a new iterative algorithm which is convergent to the signal faster than the old one. In section four, if the form of φ is simple and the sampling set is the integer set Z, then a direct reconstruction formula is presented.

在第二节，我们通过计算对偶框架来重建信号，这与传统的计算对偶框架方法不一样，该算法是通过计算一个矩阵的逆矩阵来计算对偶框架，当矩阵是奇异矩阵时，可通过奇异值分解方法或截断奇异值分解算法来求矩阵的广义逆矩阵。

In view of these, the second part of this paper presents two sufficient conditions and two mixed type duals for the generalized fractional programming only under-convexity assumptions. These sufficient conditions apply to a broader class of mathematical programming problems.The results about weak duality, strong duality and strictly reverse duality arealso presented under more suitable conditions.

鉴于此，本文的第二部分，我们仅在函数—凸性假设下，给出了广义分式规划的二个最优性充分条件，这些充分条件较文献中的相关的条件有更广泛的适用性；同时还给出了混合型对偶，并且在适当的条件下，给出了相应的弱对偶定理、强对偶定理，以及严格逆对偶定理。

With Death guitarist Schuldiner adopting vocal duties, the band made a major impact on the scene.

随着死亡的吉他手Schuldiner接受主唱的职务，乐队在现实中树立了重要的影响。

But he could still end up breakfasting on Swiss-government issue muesli because all six are accused of nicking around 45 million pounds they should have paid to FIFA.

不过他最后仍有可能沦为瑞士政府&议事餐桌&上的一道早餐，因为这所有六个人都被指控把本应支付给国际足联的大约4500万英镑骗了个精光。