Algebraic geometry and its applications by Chaumine J., et al. (eds.)

By Chaumine J., et al. (eds.)

This quantity covers many issues together with quantity conception, Boolean features, combinatorial geometry, and algorithms over finite fields. This ebook includes many attention-grabbing theoretical and applicated new effects and surveys awarded by means of the simplest experts in those components, corresponding to new effects on Serre's questions, answering a query in his letter to best; new effects on cryptographic functions of the discrete logarithm challenge concerning elliptic curves and hyperellyptic curves, together with computation of the discrete logarithm; new effects on functionality box towers; the development of latest periods of Boolean cryptographic capabilities; and algorithmic purposes of algebraic geometry.

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Thanks to Stephen Face for fruitful discussion and to Zachary Abel and Martin Demaine for their assistance with references. References 1. : Hinged dissections exist. Discrete Comput. Geom. 47(1), 150–186 (2012) 2. : Convex developments of a regular tetrahedron. Comput. Geom. Theory Appl. 34(1), 2–10 (2006) 3. : Wrapping a cube. Teach. Math. Appl. 16(3), 95–100 (1997) 4. : Problem 10716: A cubical gift. Am. Math. Monthly 108(1), 81–82 (2001) On Wrapping Spheres and Cubes with Rectangular Paper 43 5.

On the flat paper, all of the points in S are within y/2 of the major path P . Because f is contractive, all of these distances can only decrease when S is mapped onto the sphere. y dx x Fig. 2. x × y stadium, dashed major path. Fig. 3. Extension of a stadium by dx. On Wrapping Spheres and Cubes with Rectangular Paper 35 Proposition 2. An x × y stadium of flat paper mapped onto an R-sphere may occupy no more surface area than A(x, y) = 2R πR − πR cos y y + x sin . 2R 2R Proof. To bound A(x, y) we will first establish A(0, y) and then bound the derivative dA/dx.

Each cut can be performed in O(log n)-time by using the ray-shooting algorithm given in [1]. It is worth to note that Theorem 4 is based on Algorithm 1, and now this algorithm performs, as first step, a non-optimal decomposition of P into convex octilinear components. Unfortunately, Theorem 2 implies that such a convex decomposition cannot be performed efficiently. 7 Conclusion and Future Work In this work we have introduced the opd-tr problem, which consists in finding the minimal decomposition of an octilinear polygon with holes into basic components (octilinear triangles and rectangles).

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