Related papers: Automated Proofs in Geometry : Computing Upper Bou…

Variance parameter estimation in linear mixed models is a challenge for many classical nonlinear optimization algorithms due to the positive-definiteness constraint of the random effects covariance matrix. We take a completely novel view on…

The computation of amoebas has been a challenging open problem for the last dozen years. The most natural approach, namely to compute an amoeba via its boundary, has not been practical so far since only a superset of the boundary, the…

We present a computational approach to general hyperelliptic Riemann surfaces in Weierstrass normal form. The surface is either given by a list of the branch points, the coefficients of the defining polynomial or a system of cuts for the…

We study how the supporting hyperplanes produced by the projection process can complement the method of alternating projections and its variants for the convex set intersection problem. For the problem of finding the closest point in the…

We present a method for computing the topological entropy of one-dimensional maps. As an approximation scheme, the algorithm converges rapidly and provides both upper and lower bounds.

We give a new complexity bound for calculating the complex dimension of an algebraic set. Our algorithm is completely deterministic and approaches the best recent randomized complexity bounds. We also present some new, significantly sharper…

In this paper, we prove the existence of a classical solution to a Neumann boundary problem for Hessian equations in uniformly convex domain. The methods depend upon the established of a priori derivative estimates up to second order. So we…

New algorithms are devised for finding the maxima of multidimensional point samples, one of the very first problems studied in computational geometry. The algorithms are very simple and easily coded and modified for practical needs. The…

Recall that the Hilbert (Riemann-Hilbert) boundary value problem was recently solved in \cite{R1} for arbitrary measurable coefficients and for arbitrary measurable boundary data in terms of nontangential limits and principal asymptotic…

This survey focuses on the computational complexity of some of the fundamental decision problems in 3-manifold theory. The article discusses the wide variety of tools that are used to tackle these problems, including normal and almost…

Automata provide a decision procedure for Presburger arithmetic. However, until now only crude lower and upper bounds were known on the sizes of the automata produced by this approach. In this paper, we prove an upper bound on the the…

We obtain some rigidity results for overdetermined boundary value problems for singular solutions in bounded domains.

This paper uses mathematics to analyze the challenges of geometrically noisy environments on triangulation. Given widely accepted algorithmic triangulation methods, such as O (n ln n) or a simpler O (n^3) method, we can mathematically prove…

We present deterministic techniques for computing upper and lower bounds on marginal probabilities in sigmoid and noisy-OR networks. These techniques become useful when the size of the network (or clique size) precludes exact computations.…

We describe an algorithm for finding sharp upper bounds for the total Betti numbers of a saturated ideal given certain constraints on its Hilbert function. This algorithm is implemented in the Macaulay2 package, MaxBettiNumbers, along with…

We give uniform upper bounds for the number of integral points of bounded height on affine hypersurfaces, which generalise earlier results of Browning,Heath-Brown and the author.

We define modular equations in the setting of PEL Shimura varieties as equations describing Hecke correspondences, and prove upper bounds on their degrees and heights. This extends known results about elliptic modular polynomials, and…

In this paper, we study the computation of curvatures at the singular points of algebraic curves and surfaces. The idea is to convert the problem to compute the curvatures of the corresponding regular parametric curves and surfaces, which…

We propose a new numerical domain decomposition method for solving elliptic equations on compact Riemannian manifolds. One advantage of this method is its ability to bypass the need for global triangulations or grids on the manifolds.…

We consider the problem of estimating the spectral norm of a matrix using only matrix-vector products. We propose a new Counterbalance estimator that provides upper bounds on the norm and derive probabilistic guarantees on its…