Wen-tsün Wu’s scientific contributions

... Their works are collectively referred to as the Smale-Hirsch theory today. Based on this widely applicable theory, in 1964, Wu studied immersions of an arbitrary oriented 3-manifold into the 5-space [Wu64]. Precisely, he gave a complete invariant of immersions up to regular homotopy, which consists of two components. ...

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Regular homotopy classes of links of simple singularities and immersions associated with their Dynkin diagrams

... These problems are all concerned with the stability of Nash equilibria for differential games with respect to the right-hand side functions. To this end, Wu and Jiang [11] put forward the concept of essential equilibria for n-person noncooperative games, which is widely applied to study the stability of Nash equilibria in various games including differential games with respect to linear and nonlinear state functions (see, for instance, [9,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28]). For instance, by means of considering some proper conditions and using the set-valued analysis theory, Yu and Peng [9] studied the stability of equilibria against the perturbation of the right-hand side functions for noncooperative differential games and further characterized that the differential games whose equilibria are all stable form a dense residual set, and so every differential game can be approximated arbitrarily by a sequence of essential differential games. ...

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Ordering stability of Nash equilibria for a class of differential games

... It is interesting to notice that the usual definition of the fixed point property does not involve any " convex " sets but, nevertheless, the techniques using abstract convexities prove very powerful. In [ 171 Leray proved a fixed point theorem which has been extended by Wu Wen-tsiin [24] to the case of relations; it says that, given a compact space X, convexoidal in the sense of Leray [17] , every upper semicontinuous relation @ G Xx X with nonzero index A(@) and such that all sections Q(x) : = {x' 1 ( x, x') E @} have the same Tech-Alexander cohomologies as the point, must have a fixed element x0 E @(x0). A part of the present paper can be regarded as an elementary (nonalgebraic) retinement and continuation of this subject; another part can be seen as an analysis of special cases of convexoidality. ...

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The Kakutani property and the fixed point property of topological spaces with abstract convexity

... In what follows consider n > 1. The " hard " Whitney embedding theorem states that a differentiable manifold of dimension n embeds in R 2n [Whitney, 1944, p. 236] and with the orientability assumption it was known before that the result could be improved to M n → R 2n−1 for n = 4 ( n = 2 by the classification theorem for surfaces, n = 3 by Hirsch [Hirsch, 1961], n ≥ 5 by [Haefliger and Hirsch, 1963; Massey, 1960; Massey, 1962; Massey and Peterson, 1963; Wu, 1963]). The case n = 4 being the last unresolved one. ...

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All smooth orientable 4-dimensional manifolds embed in R^7

... However it did not generate classic, readable, synthetic construction proofs. In her PhD thesis [4], Marinković describes how theorem provers, based on algebraic methods such as Wu's method [5] and Gröbner basis method [6], and semi-synthetic methods such as area method [7], integrated within GCLC tool [8] and OpenGeoProver [9], could be employed to check the construction correctness. The problem with this approach is that generated proofs are not human-readable. ...

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Readable automated proofs of ruler and compass constructions