Finding equilibria of the finite size Kuramoto model amounts to solving a nonlinear system equations, which is an important yet challenging problem. We translate this into algebraic geometry problem and use numerical methods find all for various choices coupling constants K, natural frequencies, on different graphs. note that even modest sizes (N ~ 10-20), number already more than 100,000. analyze stability each computed equilibrium as well configuration angles. Our exploration landscape...

In 2013, Abo and Wan studied the analogue of Waring’s problem for systems skew-symmetric forms identified several defective systems. Of particular interest is when a certain secant variety Segre-Grassmann expected to fill natural ambient space, but actually hypersurface. Algorithms implemented in Bertini [6] are used determine degrees these hypersurfaces, representation-theoretic descriptions their equations given. We answer ([3], Problem 6.5), confirm speculation that each member an...

When fixing a covariant gauge, most popularly the Landau on lattice, one encounters Neuberger $0/0$ problem, which prevents from formulating Becchi--Rouet--Stora--Tyutin symmetry lattice. Following interpretation of this problem in terms Witten-type topological field theory and using recently developed Morse for orbifolds, we propose modification lattice gauge via orbifolding gauge-fixing group manifold show that circumvents orbit-dependence issue hence can be viable candidate evading...

In 2013, Abo and Wan studied the analogue of Waring's problem for systems skew-symmetric forms identified several defective systems. Of particular interest is when a certain secant variety Segre-Grassmann expected to fill natural ambient space, but actually hypersurface. Algorithms implemented in Bertini are used determine degrees these hypersurfaces, representation-theoretic descriptions their equations given. We answer Problem 6.5 [Abo-Wan2013], confirm speculation that each member an...

A computationally challenging classical elimination theory problem is to compute polynomials which vanish on the set of tensors a given rank. By moving away from computing via pseudowitness sets numerical theory, we develop computational methods for ranks and border along with decompositions. More generally, present our approach using joins any collection irreducible nondegenerate projective varieties $X_1,\ldots,X_k\subset\mathbb{P}^N$ defined over $\mathbb{C}$. After $\mathbb{C}$, also...

Minimizing the Euclidean distance (ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> -norm) from a given point to solution set of system polynomial equations can be accomplished via critical techniques. This article extends techniques minimization with respect Hamming xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> -"norm") and taxicab xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> -norm). Numerical algebraic geometric are derived...