By Stancho Dimiev, Kouei Sekigawa

This quantity offers the state-of-the-art contributions to the 7th foreign Workshop on advanced constructions and Vector Fields, which was once geared up as a continuation of the excessive profitable previous workshops on related study. the quantity contains works treating formidable subject matters in differential geometry, mathematical physics and expertise reminiscent of Bezier curves in house kinds, capability and catastrophy of a cleaning soap movie, computer-assisted experiences of logistic maps, and robotics.

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**Sample text**

This step is carried over by means of approximation by solutions of the homogeneous problem (Lemma 11). A variation of Lemma 2 is the following: 24 L. CAFFARELLI, ESTIMATES AND GEOMETRY OF THE MONGE-AMPERE EQUATION Lemma 9. Let u be an element of S(f) in fl D B1. Assume that DI nQr, (x1) 36 0 and that Ilf II L"(B,) 5 8 (small enough), then there exist M and u, depending on A,,\, r1i r2,17, such that IDM nQr,(xi)I > 1z1Qr,(x1)I > 0. Proof. Let xo E D1 n Qr, (x1). Subtracting from u a linear function, we may suppose that the paraboloid at xo is 1-1x12.

Corollary 2. Assume that u is strictly convex, then u has a unique supporting plane at each point. Proof. If not we may assume that a) axn (a>0), b) u(0) = 0, c) u(-ten) = 0(t) (t > 0). We now consider the auxiliary function tl,,r =u-T(Xn+a). From the strict convexity of u, the set u,,r < 0 becomes compactly contained in the domain of definition of u for a, T positive and small. Also for T < a, u,,r attains its minimum at x = 0. Finally, we estimate the location of the two supporting planes to the set u,,r <0 of the form HI = {xn = C-},113 = {xn = C+}.

Indeed, consider a sequence uk for which sup h112(X - Xo) > 1 - 1/k. x#xo hl (X - Xo) From Lemma 1, Xo, {uk = 1/21 and {uk = 1} stay uniformly away from each other. In particular 0 < CI < C2. C1 IX -X01