Suppose that f is a real univariate polynomial of degree D with exactly 4 monomial terms. We present a deterministic algorithm of complexity polynomial in logD that, for most inputs, counts the number of real roots of f . The best previous algorithms have complexity super-linear in D. We also discuss connections to sums of squares and Adiscriminants, including explicit obstructions to expressin...

It is widely believed that Krylov subspace iterative methods are better than Chebyshev semi-iterative methods. When the solution of a linear system with a symmetric and positive definite coefficient matrix is required then the Conjugate Gradient method will compute the optimal approximate solution from the appropriate Krylov subspace, that is, it will implicitly compute the optimal polynomial. ...

where S(n) is the n dimensional unit sphere in IR. This is an instance of semi-definite programming, which is a special case of convex programming, which can be solved in polynomial time (solved here means approximated within arbitrary constant in polynomial time). Observe that (P) is a relaxation of (Q), and as such the optimal solution of (P) has value larger than the optimal value of (Q). Th...

Community detection refers to the problem of clustering nodes a network (either graph or hypergrah) into groups. Various algorithms are available for community and all these methods apply uncensored networks. In practice, may has censored (or missing) values it is shown that have non-negligible effect on structural properties network. this paper, we study in $m$-uniform hypergraph from informat...

This manuscript concerns two definite integrals that could be connected to the Bose-Einstein and Fermi-Dirac functions in integrands, separately, with numerators slightly modified a difference expressions contain Fourier kernel multiplied by polynomial its complex conjugate. In this work, we use our contour integral method derive these integrals, which are given ∫0∞ie−imx(log(a)−ix)k−eimx(log(a...

In his solution of Hilbert's 17th problem Artin showed that any positive definite polynomial in several variables can be written as the quotient two sums squares. Later Reznick denominator Artin's result always chosen an $N$-th power squared norm and gave explicit bounds on $N$. By using concepts from quantum information theory (such partial traces, optimal cloning maps, identity due to Chiribe...

The goal of this paper is to present a systematic method compute reference dependent positively invariant sets for systems subject constraints. To end, we first characterize these as level Lyapunov functions. Based on characterization and using Sum Squares theory, provide polynomial certificate the existence such sets. Subsequently, through some algebraic manipulations, express in terms Semi-De...

Neural Networks have been widely used to solve Partial Differential Equations. These methods require approximate definite integrals using quadrature rules. Here, we illustrate via 1D numerical examples the problems that may arise in these applications and propose different alternatives overcome them, namely: Monte Carlo methods, adaptive integration, polynomial approximations of Network output,...