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Emiris I.Z.,National and Kapodistrian University of Athens | Fisikopoulos V.,National and Kapodistrian University of Athens | Konaxis C.,University of Crete | Penaranda L.,Impa Instituto Nacional Of Matematica Pura E Aplicada
International Journal of Computational Geometry and Applications | Year: 2013

We design an algorithm to compute the Newton polytope of the resultant, known as resultant polytope, or its orthogonal projection along a given direction. The resultant is fundamental in algebraic elimination, optimization, and geometric modeling. Our algorithm exactly computes vertex-and halfspace-representations of the polytope using an oracle producing resultant vertices in a given direction, thus avoiding walking on the polytope whose dimension is α-n-1, where the input consists of α points in ℤn. Our approach is output-sensitive as it makes one oracle call per vertex and per facet. It extends to any polytope whose oracle-based definition is advantageous, such as the secondary and discriminant polytopes. Our publicly available implementation uses the experimental CGAL package triangulation. Our method computes 5-, 6-and 7-dimensional polytopes with 35K, 23K and 500 vertices, respectively, within 2hrs, and the Newton polytopes of many important surface equations encountered in geometric modeling in < 1sec, whereas the corresponding secondary polytopes are intractable. It is faster than tropical geometry software up to dimension 5 or 6. Hashing determinantal predicates accelerates execution up to 100 times. One variant computes inner and outer approximations with, respectively, 90% and 105% of the true volume, up to 25 times faster. © 2013 World Scientific Publishing Company.

Finster F.,University of Regensburg | Reintjes M.,Impa Instituto Nacional Of Matematica Pura E Aplicada
Advances in Theoretical and Mathematical Physics | Year: 2015

We give a functional analytic construction of the fermionic projector on a globally hyperbolic Lorentzian manifold of finite lifetime. The integral kernel of the fermionic projector is represented by a two-point distribution on the manifold. By introducing an ultraviolet regularization, we get to the framework of causal fermion systems. The connection to the negative-energy solutions of the Dirac equation and to the WKB approximation is explained and quantified by a detailed analysis of closed Friedmann-Robertson-Walker universes.

Reintjes M.,Impa Instituto Nacional Of Matematica Pura E Aplicada | Temple B.,University of California at Davis
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences | Year: 2015

We give a constructive proof that coordinate transformations exist which raise the regularity of the gravitational metric tensor from C0,1 to C1,1 in a neighbourhood of points of shock wave collision in general relativity. The proof applies to collisions between shock waves coming from different characteristic families, in spherically symmetric spacetimes. Our result here implies that spacetime is locally inertial and corrects an error in our earlier Proc. R. Soc. A publication, which led us to the false conclusion that such coordinate transformations, which smooth the metric to C1,1, cannot exist. Thus, our result implies that regularity singularities (a type of mild singularity introduced in our Proc. R. Soc. A paper) do not exist at points of interacting shock waves from different families in spherically symmetric spacetimes. Our result generalizes Israel's celebrated 1966 paper to the case of such shock wave interactions but our proof strategy differs fundamentally from that used by Israel and is an extension of the strategy outlined in our original Proc. R. Soc. A publication. Whether regularity singularities exist in more complicated shock wave solutions of the Einstein-Euler equations remains open. © 2015 The Authors.

Izmailov A.F.,Moscow State University | Izmailov A.F.,Peoples Friendship University of Russia | Izmailov A.F.,Tambov State University | Solodov M.V.,Impa Instituto Nacional Of Matematica Pura E Aplicada
Optimization Methods and Software | Year: 2016

For the sequential quadratic programming method (SQP), we show that close to a solution satisfying the same assumptions that are required for its local quadratic convergence (namely uniqueness of the Lagrange multipliers and the second-order sufficient optimality condition), the direction given by the SQP subproblem using the Hessian of the Lagrangian is a descent direction for the standard (Formula presented.)-penalty function. We emphasize that this property is not straightforward at all, because the Hessian of the Lagrangian need not be positive definite under these assumptions or, in fact, under any other reasonable set of assumptions. In particular, this descent property was not known previously, under any assumptions (even including the stronger linear independence constraint qualification, strict complementarity, etc.). We also check the property in question by experiments on nonconvex problems from the Hock–Schittkowski test collection for a model algorithm. While to propose any new and complete SQP algorithm is not our goal here, our experiments confirm that the descent condition, and a model method based on it, work as expected. This indicates that the new theoretical findings that we report might be useful for full/practical SQP implementations which employ second derivatives and linesearch for the (Formula presented.)-penalty function. In particular, our results imply that in SQP methods where using subproblems without Hessian modifications is an option, this option has a solid theoretical justification at least on late iterations. © 2016 Informa UK Limited, trading as Taylor & Francis Group

Vital Brazil E.,University of Calgary | Amorim R.,University of Calgary | Costa Sousa M.,University of Calgary | Velho L.,Impa Instituto Nacional Of Matematica Pura E Aplicada | De Figueiredo L.H.,Impa Instituto Nacional Of Matematica Pura E Aplicada
Computers and Graphics (Pergamon) | Year: 2015

We present a theoretical approach to the problem of sketch-based surface modeling (SBSM) and introduce a framework for SBSM systems based on adaptive meshes. The main advantage of this approach is a clear separation between the modeling operators and the final representation, thus enabling the creation of SBSM systems suited to specific domains with different demands. To support the proposed approach we present two different sketch-based modeling systems built using this framework. The first one has the capability to control local and global changes to the model; the second one follows geological data constraints. To build the first system that provides the user with control of local modifications we developed a mathematical theory of vertex label and atlas structure for adaptive meshes based on stellar operators. © 2015 Elsevier Ltd. All rights reserved.

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