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).