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In many commercial and academic settings, numerical solvers fail to achieve their theoretical performance levels due to issues in the system definition, parameterization, and even implementation. We propose a pair of methods for detecting…
We count integer points on bihomogeneous varieties using the Hardy-Littlewood method. The main novelty lies in using the structure of bihomogeneous equations to obtain asymptotics in generically fewer variables than would be necessary in…
We develop a numerical approach to cohomology. Essentially, vector spaces and linear maps are replaced by real numbers, which represent dimensions of vector spaces and ranks of linear maps. We use this to refine ideas of Van der Geer and…
Let $X$ be a complex manifold and let $g$ be a polyhedral metric on it inducing its topology. We say that $g$ is a polyhedral K\"ahler (PK) metric on $X$ if it is K\"ahler outside its singular set. The local geometry of PK metrics is…
We extend a recently developed numerical code to obtain stationary, axisymmetric solutions that describe rotating black hole spacetimes in a wide class of modified theories of gravity. The code utilizes a relaxed Newton-Raphson method to…
For families of Hamiltonians defined by parts that are local, the most general definition of a symmetry algebra is the commutant algebra, i.e., the algebra of operators that commute with each local part. Thinking about symmetry algebras as…
In this paper we proposed the homotopy approach for solving the nonlinear Balitsky-Kovchegov (BK) evolution equation with running QCD coupling. The approach consists of two steps. First, is the analytic solution to the nonlinear evolution…
Modern categories of spectra such as that of Elmendorf et al equipped with strictly symmetric monoidal smash products allows the introduction of symmetric monoids providing a new way to study highly coherent commutative ring spectra. These…
Significant advances in numerical simulations of black-hole binaries have recently been achieved using the puncture method. We examine how and why this method works by evolving a single black hole. The coordinate singularity and hence the…
In the present paper we show a dichotomy theorem for the complexity of polynomial evaluation. We associate to each graph H a polynomial that encodes all graphs of a fixed size homomorphic to H. We show that this family is computable by…
To solve boundary integral equations for potential problems using collocation Boundary Element Method (BEM) on smooth curved 3D geometries, an analytical singularity extraction technique is employed. By adopting the isoparametric approach,…
These pages contain a short overview on the state of the art of efficient numerical analysis methods that solve systems of multivariate polynomial equations. We focus on the work of Steve Smale who initiated this research framework, and on…
We study the modular resolution method using new tools called polynumbers and polyseries, introduced by Prof. Wildberger N.J. We try to prove an equivalence theorem of the existence and the uniqueness of the solutions of the modular…
Via a generalization of the pseudospectral method for numerical solution of differential equations, a family of nonlinear algebraic identities satisfied by the zeros of a wide class of orthogonal polynomials is derived. The generalization…
We prove a conjecture of Bahri, Bendersky, Cohen and Gitler: if K is a shifted simplicial complex on n vertices, X_1,..., X_n are spaces and CX_i is the cone on X_i, then the polyhedral product determined by K and the pairs (CX_i,X_i) is…
An ideal of a local polynomial ring can be described by calculating a standard basis with respect to a local monomial ordering. However standard basis algorithms are not numerically stable. Instead we can describe the ideal numerically by…
Homotopy Lie algebras are a generalization of differential graded Lie algebras encoding both the kinematics and dynamics of a given field theory. Focusing on kinematics, we show that these algebras provide a natural framework for the…
An algorithmic computation of the set of unpointed stable homotopy classes of equivariant fibrewise maps was described in a recent paper of the author and his collaborators. In the present paper, we describe a simplification of this…
Differential calculi are obtained for quantum homogeneous spaces by extending Woronowicz' approach to the present context. Representation theoretical properties of the differential calculi are investigated. Connections on quantum…
We survey results on the distribution of zeros of random polynomials and of random holomorphic sections of line bundles, especially for large classes of probability measures on the spaces of holomorphic sections. We provide furthermore some…