By Takuji Arai, Takamasa Suzuki (auth.), Shigeo Kusuoka, Toru Maruyama (eds.)

A lot of financial difficulties will be formulated as limited optimizations and equilibration in their strategies. a variety of mathematical theories were delivering economists with critical machineries for those difficulties coming up in financial idea. Conversely, mathematicians were inspired through a variety of mathematical problems raised by means of fiscal theories. The sequence is designed to compile these mathematicians who're heavily drawn to getting new tough stimuli from monetary theories with these economists who're looking potent mathematical instruments for his or her research.

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The image and preimage of a semi-algebraic set under semi-algebraic (set-valued) mapping is semi-algebraic. Composition of semi-algebraic (set-valued) mappings is a semi-algebraic mapping. The marginal function inf f (x, y) is semi-algebraic if so is f . x For our discussions the most important results about semi-algebraic sets and mappings is a deep stratification theorem proved in Łojasiewic in 1963 (see [23]). ) Before stating the theorem we recall that a Whitney stratification of a set Q ⊂ IRn is a partition of Q into a finite number of smooth manifolds Mi (strata) which meet each other in a certain regular way, namely (a) If xn ∈ Mj converge to an x ∈ Mi (i = j ) and h ∈ Tx Mi (the tangent space to Mi at x), then there are hn ∈ Txn Mj converging to h.

We show ∗ A preliminary version of this work was presented at the Workshop on Mathe- matical Economics held in Keio University, Tokyo during November 13–15, 2009. The authors are grateful to Professor Toru Maruyama for his invitation, and to him and Professors Alexander Ioffe, Alejandro Jofre, Boris Mordukhovich and Haru Takahashi for discussion and encouragement. Adriana Piazza gratefully acknowledges the financial support of Programa Basal PFB 03, Centro de Modelamiento Matem´atico (CMM), Universidad de Chile and that of FONDECYT under project 11090254.

But the latter cannot happen. Indeed, we can view Si as the graph of a set-valued mapping from IR m into IRk . This mapping is regular at (x, w) by (c) and, as we have seen, q = 0, so Theorem 6 excludes such a possibility. This completes the proof. We are now ready to prove a semi-algebraic version of Theorem 2. Theorem 13. Let F : IR m × IRk ⇒ IR n be a semi-algebraic set-valued mapping with locally closed graph, and let y be a regular value of F . Then for all w ∈ IR k , with a possible exception of a semi-algebraic set of dimension strictly smaller than k, y is a regular value of the partial mapping Fw : x → F (x, w).