查询词典 strictly convex function

At the beginning of this thesis, the author gives the definition and the equivalent definition of convex function, and then proves the equivalent relationship between them. Secondly the author proposes the decision theorem of convex function which provides a judgment basis of whether a function is a convex function. Thirdly the author summarizes and proves the convex function's operational, basic, differential and integral property. Finally the author proves several famous convex function inequalities, such as Jensen inequality, Holder inequality, Cauchy inequality. The author also provides the application of these inequalities and illustrates the importance of convex function's basic inequality and integral property in the proving process.

本文开始给出了凸函数的定义及等价定义，并证明了它们之间的等价关系；接着提出了凸函数的判定定理，对一个函数是否是凸函数提供判断依据；然后对凸函数的运算性质、基本性质、微分性质、积分性质四个方面的性质进行了总结，并给予了证明；最后证明了凸函数的几个著名不等式詹森不等式、赫尔德不等式、柯西不等式，给出了这几个不等式的一些应用实例，并举例说明凸函数的基本性质和积分性质在不等式证明过程中的重要作用。

At the beginning of this thesis, the author gives the definition and the equivalent definition of convex function, and then proves the equivalent relationship between them. Secondly the author proposes the decision theorem of convex function which provides a judgment basis of whether a function is a convex function. Thirdly the author summarizes and proves the convex function's operational ,basic , differential and integral property. Finally the author proves several famous convex function inequalities, such as Jensen inequality, Holder inequality, Cauchy inequality and Minkowski inequality. The author also provides the application of these inequalities and illustrates the importance of convex function's basic inequality and integral property in the proving process.

本文开始给出了凸函数的定义及等价定义，并证明了它们之间的等价关系；接着提出了凸函数的判定定理，对一个函数是否是凸函数提供判断依据；然后对凸函数的运算性质、基本性质、微分性质、积分性质四个方面的性质进行了总结，并给予了证明；最后证明了凸函数的几个著名不等式詹森不等式、赫尔德不等式、柯西不等式和闵可夫斯基不等式以及这几个不等式的应用，并举例说明凸函数的基本性质和积分性质在不等式证明过程中的重要作用。

The convex function develeped so many years that convex functionals are now a class of very universal functionals.The kinetic energy in the classical is the most natural convex function.Others like entropy and so on are convex functions.The economic core is closely connected with the convex function.The western economic core question is the optimized question.The optimized theory has something to do with the convex function and the concave function.

凸函数的发展经年已相当完善其应用非常广泛，在古典力学中的动能，就是最自然直接的凸函数，其他如熵……等均是，连经济学的核心都与之有极密切相关，西方经济学的核心问题是最优化问题，最优化理论又与凸函数与凹函数密切相关。

Chapter 4 establishes the theory of convex fuzzy mappings: The concepts, such as Jensen's inequality, positively homogeneous, infimal convolution, right scalar multiplication and convex hull are introduced. The corresponding theorems are demonstrated by using the parametric representations of fuzzy numbers. In anti-fuzzy number space, the conjugate mapping of convex fuzzy mapping is concerned, and convexities of conjugate set and conjugate mapping of convex fuzzy mapping are proved. The notions of subgradient, subdifferential, differential with respect to convex fuzzy mappings are investigated, which provides the basis of the theory of fuzzy extremum problems.

在第4章中，建立了有关凸模糊映射的理论：建立了关于凸模糊映射的Jensen不等式、模糊正齐次映射、凸模糊映射的下卷积、右数乘和凸包等概念，利用模糊数的参数化表示，给出了相应的定理；在反模糊数空间，对凸模糊映射的共轭也作了探讨，证明了凸模糊映射的共轭集合和共轭映射都是凸的；最后对凸模糊映射的次梯度、次微分和微分等概念进行了研究，为模糊极值理论打下了基础。

In this paper, a varieties of concepts of compactness of sets and sunsets are introduced in a complex locally convex space, which are respectively the generalizations of the known sunsets and convex sets. By establishing the relationship between restricted p-centers of a set in a complex locally convex space and restricted Chebyshev centers of the corresponding set in the quotient space with respect to ker, characterizations and uniqueness of restricted p-centers of sets in a complex locally convex space are obtained. These results are better than any other known ones when we restrict them in real normed linear spaces and real locally convex space.

本文首先在复局部凸空间中引入了几种紧性及几种太阳集的概念，它们分别是已有紧性和凸集概念的推广；然后通过建立局部凸空间中集的限制p-中心和关于ker的商空间中对应集的限制Chebyshev中心的关系，来研究局部凸空间中集的限制p-中心问题，本文即使是在实赋范空间和在实局部凸空间中所得的结果都分别是有关已知结果的深化和推广。

Technical characteristics: concavo-convex Nano-imprint lithography processing is a common business card printing and membership card making products that decoration, he uses the Teletext match concavo-convex stempelsysteme, by mechanical action make printing surface correspond to the Office of the permanent press deformation occurred in oilysludge, in the printing surface stereo pictures and texts, in recent years with the electronic sculpture and laser engraving technique, concavo-convex stempelsysteme mismachining standard qualitative leap, given the drlling of concavo-convex stempelsysteme makes it possible for the processing of via concavo-convex Nano-imprint lithography printed matter is more true to life on the surface of the graphics and text.

工艺特点：凹凸压印加工是一种常见的制卡和会员卡制作品表明装饰工艺，他使用图文信息相互匹配的凹凸印模，通过机械作用使印品表面对应处基材发生永久性挤压变形后，在印品表面形成立体图文，近年来随着电子雕刻和激光雕刻技术的不断提高，凹凸印模的加工水平有了质的飞跃，精雕细凿的凹凸印模使得经凹凸压印加工的印品表面之图文更加惟妙惟肖。

The decreasing conditions of an energy function are investigated by the use of convex function. By utilization of the primal convex function, the conditions are analyzed under which its conjugate function minus a quadratic function is also convex.

利用凸函数的定义研究了能量函数下降的条件，根据凸函数的性质分析它的共轭函数减去二次函数之差仍为凸函数的条件。

Properties of efficient portfolios and the efficient frontier to the model are systematically analyzed. Our main results concerning the properties are: every efficient portfolio can be solved by minimizing portfolio risk under a given level of portfolio return or by maximizing portfolio return under a given level of portfolio risk; on the efficient frontier, the risk is a convex and strictly increasing function of the return and the return is a concave and strictly increasing function of the risk; the utility function on the efficient frontier can be expressed as a quasi-concave function of the risk or the return if the investor's utility function is quasi-concave.

从理论上系统地对该模型下的有效投资组合和有效前沿的性质进行了分析，结果表明：每一个有效投资组合可通过在给定期望收益水平的条件下最小化投资组合风险来获得，或者在给定风险水平的条件下最大化期望投资组合收益来获得；在有效前沿上，风险是收益的严格单调递增凸函数，收益是风险的严格单调递增凹函数；当投资者的效用函数是拟凹函数时，则有效前沿上的效用可表达成风险或收益的拟凹函数。

It is proved that strictly efficient point set of the objective space is connected when objective function is cone convex－like,and that if objective function is cone convex set－valued mapping,then strictly efficient solution set is also connected.

证明了当目标函数为锥类凸的集值映射时，其目标空间里的严有效点集是连通的；若目标函数为锥凸的集值映射时，其严有效解集也是连通的。

Systems of inqualities,the minimum or maximum of a convex function over a convex set,Lagrange multipliers,and minimax theorems are among the topic treated,as well as basic results about the structure of convex sets and continuity and differentiability of convex functions and saddle-functions.

其中对不等式系统、凸集上的凸函数的极小或极大、拉格朗日乘子和极小极大定理做了专题论述，同时对凸集的结构和连续性以及凸函数的可微性和鞍点函数的基本结果做了介绍。

Sodium citrate; calcium carbonate; calcium hydroxide; potassium bicorbonate; magnesium carbonate; ferrous sulphate; Zinc sulphate; copper sulphate; manganese sulphate; potassium iodide; sodium selenite.

柠檬酸钠；碳酸钙；氢氧化钙；碳酸氢钾；碳酸镁；硫酸亚铁；硫酸锌；硫酸铜；硫酸锰；碘化钾；亚硒酸钠。

All this to say that I'm glad filmmaking hasn't become completely ironic. I can enjoy an ironic film as much as the next person, but they usually involve a message of some sort. One that's delivered in a smirkingly clever way that requires little in the way of talent.

所有这些都是说，我很高兴电影制作没有变成完全讽刺艺术，我可以和下一个人一样享受讽刺电影的乐趣，虽然他们经常会被划归到一定级别，技术方面不需要很高的才能。