In this paper we study the distributions of number real solutions to power flow equations over varying electrical parameters. We develop a new monodromy and parameter homotopy continuation method for quickly finding all equations. apply find compare these those random polynomials. It is observed that while tend admit many fewer real-valued than bound on total complex solutions, low levels load ...

Let G be a finite cover of a closed connected transpose-stable subgroup of GL(n,R) with complexified Lie algebra g. Let K be a maximal compact subgroup of G, and assume that G and K have equal rank. We prove a translation principle for the Dirac index of virtual (g,K)-modules. As a byproduct, to each coherent family of such modules, we attach a polynomial on the dual of the compact Cartan subal...

By applying the Euler–Rayleigh method to a specific representation of Jacobi polynomials as hypergeometric functions, we obtain new bounds for their largest zeros. In particular, derive upper and lower bound $$1-x_{nn}^2(\lambda )$$ , with $$x_{nn}(\lambda being zero n-th ultraspherical polynomial $$P_n^{(\lambda )}$$ . For every fixed $$\lambda >-1/2$$ limit ratio our does not exceed 1.6. This...

The eigenvalues of a matrix A are the zeros of its characteristic polynomial fiX) = dtt[A XI]. With Hyman's method of determinant evaluation, a new homotopy continuation method, homotopy-determinant method, is developed in this paper for finding all eigenvalues of a real upper Hessenberg matrix. In contrast to other homotopy continuation methods, the homotopy-determinant method calculates eigen...

Motivated by the recently observed phenomenon of topology trivialization of potential energy landscapes (PELs) for several statistical mechanics models, we perform a numerical study of the finite-size 2-spin spherical model using both numerical polynomial homotopy continuation and a reformulation via non-Hermitian matrices. The continuation approach computes all of the complex stationary points...

An eecient implementation of Reid's multiple hypothesis tracking (MHT) algorithm is presented in which the the k-best hypotheses are determined in polynomial time using an algorithm due to Murty 13]. The MHT algorithm is then applied to several motion sequences. The MHT capabilities of track initiation, termination and continuation are demonstrated. Continuation allows the MHT to function despi...

This is a continuation of our paper " A Theory of Pfaffian Orientations I: Perfect Matchings and Permanents ". We present a new combinatorial way to compute the generating functions of T-joins and k-cuts of graphs. As a consequence, we show that the computational problem to find the maximum weight of an edge-cut is polynomially solvable for the instances (G, w) where G is a graph embedded on an...

Real-World Neuroimaging Technologies Report Title Decades of heavy investment in laboratory-based brain imaging and neuroscience have led to foundational insights into how humans sense, perceive, and interact with the external world. However, it is argued that fundamental differences between laboratory-based and naturalistic human behavior may exist. Thus, it remains unclear how well the curren...

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