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The analogy between eigenvalues and singular values has many faces. The current review brings together several examples of this analogy. One example regards the similarity between Symmetric Rayleigh Quotients and Rectangular Rayleigh Quotients. Many useful properties of eigenvalues stem are from the Courant-Fischer minimax theorem, from Weyl’s theorem, and their corollaries. Another aspect regards “rectangular” versions of these theorems. Comparing the properties of Rayleigh Quotient matrices with those of Orthogonal Quotient matrices illuminates the subject in a new light. The Orthogonal Quotients Equality is a recent result that converts Eckart-Young’s minimum norm problem into an equivalent maximum norm problem. This exposes a surprising link between the Eckart-Young theorem and Ky Fan’s maximum principle. We see that the two theorems reflect two sides of the same coin: there exists a more general maximum principle from which both theorems are easily derived. Ky Fan has used his extremum principle (on traces of matrices) to derive analog results on determinants of positive definite Rayleigh Quotients matrices. The new extremum principle extends these results to Rectangular Quotients matrices. Bringing all these topics under one roof provides new insight into the fascinating relations between eigenvalues and singular values.

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KKM and KY fan theorems in modular function spaces
Khamsi Mohamed,Latif Abdul,Al-Sulami Hamid
Fixed Point Theory and Applications , 2011,

Abstract: In modular function spaces, we introduce Knaster-Kuratowski-Mazurkiewicz mappings (in short KKM-mappings) and prove an analogue to Ky Fan s fixed point theorem. 2010 Mathematics Subject Classification: Primary 46B20, 47H09; Secondary 47H10.

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关于第二类s-凸函数的注记
A Note on s2-Convex Functions [PDF]
李耀文, 张沂昀
Operations Research and Fuzziology (ORF) , 2016, DOI: 10.12677/ORF.2016.61002

Abstract:

本文中，我们研究了第二类s-凸函数，把凸函数的子母定理推广到s-凸函数的相应形式，由此证明了Ky-Fan型不等式，Milne型不等式。
In this paper we study the s-Convex Functions of the second type. We show that a master theorem holds for the s-Convex Functions, and we then apply it to obtain Ky-Fan type inequalities and Milne type inequalities.

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Bounding Inequalities for Eigenvalues of Principal Submatrices [PDF]
Achiya Dax
Advances in Linear Algebra & Matrix Theory (ALAMT) , 2019, DOI: 10.4236/alamt.2019.92002

Abstract: Ky Fan trace theorems and the interlacing theorems of Cauchy and Poincaré are important observations that characterize Hermitian matrices. In this note, we introduce a new type of inequalities which extend these theorems. The new inequalities are obtained from the old ones by replacing eigenvalues and diagonal entries with their moduli. This modification yields effective bounding inequalities which are valid on a larger range of matrices.

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BOUNDED RATIONALITY AND STABILITY OF SOLUTIONS OF SOME EQUILIBRIUM PROBLEMS
几类考虑有限理性平衡问题解的稳定性
YU Jian,
俞建
系统科学与数学 , 2009,

Abstract: In this paper, first rationality functions for Ky Fan's points problems are defined, and then it is proven that most of the Ky Fan's points problems (in the sense of Baire category) are structurally stable and robust to $\varepsilon$-equilibria. Finally, as applications, the stability results on Nash equilibrium problems and variational inequality problems are given.

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Maximum Principles for Normal Matrices [PDF]
Achiya Dax
Advances in Linear Algebra & Matrix Theory (ALAMT) , 2019, DOI: 10.4236/alamt.2019.93005

Abstract: Ky Fan maximum principle is a well-known observation about traces of certain hermitian matrices. In this note, we derive a powerful extension of this claim. The extension is achieved in three ways. First, traces are replaced with norms of diagonal matrices, and any unitarily invariant norm can be used. Second, hermitian matrices are replaced by normal matrices, so the rule applies to a larger class of matrices. Third, diagonal entries can be replaced with eigenvalues and singular values. It is shown that the new maximum principle is closely related to the problem of approximating one matrix by another matrix of a lower rank.

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多重拓扑下Fan Ky点的通有稳定性
The Generic Stability of Fixed Point and Fan Ky Point in Diverse Topology
黄辉,左勇华,卢美华
- , 2017,

Abstract: 建立集合族空间,讨论了公共元的通有稳定性,得到了闭集族空间上的交运算在Hausdorff 拓扑下的上半连续性.在2种拓扑结构下研究了Fan Ky点的通有稳定性,显示了集族空间交运算方法具有良好的适应性.
A family-of-set space is established.And its common elements’ generic stability is studied.The upper semi-continuity of operation of sets’ intersection in family-of-closed-set space is obtained.And generic stability of Fan Ky point is studied in diverse topology.It has good applicability of the method of sets’ intersection in family-of-closed-set space

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On a Simpler, Much More General and Truly Marvellous Proof of Fermat’s Last Theorem (I) [PDF]
Golden Gadzirayi Nyambuya
Advances in Pure Mathematics (APM) , 2016, DOI: 10.4236/apm.2016.61001

Abstract: English mathematics Professor, Sir Andrew John Wiles of the University of Cambridge finally and conclusively proved in 1995 Fermat’s Last Theorem which had for 358 years notoriously resisted all gallant and spirited efforts to prove it even by three of the greatest mathematicians of all time—such as Euler, Laplace and Gauss. Sir Professor Andrew Wiles’s proof employed very advanced mathematical tools and methods that were not at all available in the known World during Fermat’s days. Given that Fermat claimed to have had the “truly marvellous” proof, this fact that the proof only came after 358 years of repeated failures by many notable mathematicians and that the proof came from mathematical tools and methods which are far ahead of Fermat’s time, has led many to doubt that Fermat actually did possess the “truly marvellous” proof which he claimed to have had. In this short reading, via elementary arithmetic methods, we demonstrate conclusively that Fermat’s Last Theorem actually yields to our efforts to prove it.

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Existence and Uniqueness of Solution to Semilinear Fractional Elliptic Equation [PDF]
Shangjian Liu
Journal of Applied Mathematics and Physics (JAMP) , 2019, DOI: 10.4236/jamp.2019.71017

Abstract: In this work, we study the following problem. $\"\"$ , where $\"\"$ ？is the fractional Laplacian and Ω？is a bounded domain in R^N？with Lipschitz boundary. g: R→R？is an increasing locally Lipschitz continuous function. and f∈L^m(Ω), $\"\"$ . We use Stampacchia’s theorem to study existence of the solution u

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不确定性下非合作博弈强Nash均衡的存在性
张会娟,张强
控制与决策 , 2010,

Abstract: 在已知不确定参数变化范围的假设下,研究了非合作博弈强Nash均衡的存在性问题.基于经典非合作博弈的强Berge均衡及帕雷托均衡的概念,结合非合作博弈NS均衡,定义了不确定性下非合作博弈的帕雷托强Berge和强Nash均衡的概念,并借助KyFan不等式证明其存在性.最后利用算例验证了其可行性和有效性.

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