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Consider a real-valued function that can only be observed with stochastic noise at a finite set of design points within a Euclidean space. We wish to determine whether there exists a convex function that goes through the true function…

To an ideal in $\mathbb{C}[x,y]$ one can associate a topological zeta function. This is an extension of the topological zeta function associated to one polynomial. But in this case we use a principalization of the ideal instead of an…

Computing the closed convex envelope or biconjugate is the core operation that bridges the domain of nonconvex with convex analysis. We focus here on computing the conjugate of a bivariate piecewise quadratic function defined over a…

In the field of radial basis functions mathematicians have been endeavouring to find infinitely differentiable and compactly supported radial functions. This kind of functions are extremely important for some reasons. First, its…

In this paper we present a novel technique based on deep reinforcement learning that allows for numerical analytic continuation of integrals that are often encountered in one-loop diagrams in quantum field theory. In order to extract…

It is still a challenging task of today to recognize the type of a given algebraic surface which is described only by its implicit representation. In~this paper we will investigate in more detail the case of canal surfaces that are often…

These lecture notes present a computation driven pathway from classical complex analysis to the theory of compact Riemann surfaces and their connections to algebraic geometry. The exposition follows a compute first then abstract philosophy,…

Classical algorithms in numerical analysis for numerical integration (quadrature/cubature) follow the principle of approximate and integrate: the integrand is approximated by a simple function (e.g. a polynomial), which is then integrated…

We construct and apply the classic balayage (sweeping out) of measures and subharmonic functions on closed system of rays in the complex plane with vertex at the origin, including measures and subharmonic functions and infinite order. The…

The circle method has been successfully used over the last century to study rational points on hypersurfaces. More recently, a version of the method over function fields, combined with spreading out techniques, has led to a range of results…

This paper proposes a method for computing the visible occluding contours of subdivision surfaces. The paper first introduces new theory for contour visibility of smooth surfaces. Necessary and sufficient conditions are introduced for when…

An algorithm for computing an analytic function of a matrix $A$ is described. The algorithm is intended for the case where $A$ has some close eigenvalues, and clusters (subsets) of close eigenvalues are separated from each other. This…

We are interested in the fast computation of the exact value of integrals of polynomial functions over convex polyhedra. We present speed ups and extensions of the algorithms presented in previous work. We present the new software…

In our previous paper, Real Polynomials with a Complex Twist [see http://archives.math.utk.edu/ICTCM/VOL28/A040/paper.pdf], we used advancements in computer graphics that allow us to easily illustrate more complete graphs of polynomial…

Monte Carlo studies of many quantum systems face exponentially severe signal-to-noise problems. We show that noise arising from complex phase fluctuations of observables can be reduced without introducing bias using path integral contour…

Using a new technique involving integration it is possible to find the exact roots of simple functions. In this case, simple functions are defined as smooth functions having an inverse, and that inverse having an antiderivative. This…

We consider a large class of physical fields $u$ written as double inverse Fourier transforms of some functions $F$ of two complex variables. Such integrals occur very often in practice, especially in diffraction theory. Our aim is to…

The essence of Stahl-Gonchar-Rakhmanov theory of symmetric contours as applied to the multipoint Pad\'e approximants is the fact that given a germ of an algebraic function and a sequence of rational interpolants with free poles of the germ,…

We study a new class of Radon transforms defined on circular cones called the conical Radon transform. In $\mathbb{R}^3$ it maps a function to its surface integrals over circular cones, and in $\mathbb{R}^2$ it maps a function to its…

The common approach to radial distortion is by the means of polynomial approximation, which introduces distortion-specific parameters into the camera model and requires estimation of these distortion parameters. The task of estimating…