We consider a pair of collectively oscillating networks dynamical elements and optimize their internetwork coupling for efficient mutual synchronization based on the phase reduction theory developed by Nakao et al. [Chaos 28, 045103 (2018)]. The equations describing weakly coupled are reduced to equations, linear stability synchronized state between is represented as function matrix. seek optim...

A new derivation is presented for the least squares solution of the design problem of 2-D FIR filters by minimizing the Frobenius norm of the difference between the matrices of the ideal and actual frequency responses sampled at the points of a frequency grid. The mathematical approach is based on the singular value decomposition of two complex transformation matrices. Interestingly, the design...

Let L/K be a Galois extension of number fields. The problem of counting the number of prime ideals p of K with fixed Frobenius class in Gal(L/K) and norm satisfying a congruence condition is considered. We show that the square of the error term arising from the Chebotarëv Density Theorem for this problem is small “on average.” The result may be viewed as a variation on the classical Barban-Dave...

Part II. p-adic methods §3. Considerations on the differential case §4. Introduction to p-adic q-difference modules 4.1. p-adic estimates of q-binomials 4.2. The Gauss norm and the invariant χv(M) 4.3. q-analogue of the Dwork-Frobenius theorem §5. p-adic criteria for unipotent reduction 5.1. q-difference modules having unipotent reduction modulo ̟v 5.2. q-difference modules having unipotent redu...

We demonstrate the application of Widening to learning performant Bayesian Networks for use as classifiers. Widening is a framework for utilizing parallel resources and diversity to find models in a hypothesis space that are potentially better than those of a standard greedy algorithm. This work demonstrates that widened learning of Bayesian Networks, using the Frobenius Norm of the networks’ g...

In this paper we show that the left-invariant geodesic flow on the symplectic group with metric given by the Frobenius norm is an integrable system that is not contained in the Mishchenko-Fomenko class. We show that this system may be expressed as a flow on symmetric matrices and that the system is biHamiltonian. Research partially supported by the NSF. Research partially supported by the Calif...

We consider the problem of selecting the “best” subset of exactly k columns from an m× n matrix A. In particular, we present and analyze a novel two-stage algorithm that runs in O(min{mn2,m2n}) time and returns as output an m × k matrix C consisting of exactly k columns of A. In the first stage (the randomized stage), the algorithm randomly selects O(k log k) columns according to a judiciously-...

Covariance matrix plays a central role in multivariate statistical analysis. Significant advances have been made recently on developing both theory and methodology for estimating large covariance matrices. However, a minimax theory has yet been developed. In this paper we establish the optimal rates of convergence for estimating the covariance matrix under both the operator norm and Frobenius n...

Covariance matrix plays a central role in multivariate statistical analysis. Significant advances have been made recently on developing both theory and methodology for estimating large covariance matrices. However, a minimax theory has yet been developed. In this paper we establish the optimal rates of convergence for estimating the covariance matrix under both the operator norm and Frobenius n...

We present a formulation for the structured condition number and for the structured backward error for the linear system A Ax = b, when the rectangular matrix A is subjected to normwise perturbations. Perturbations on the data A and the solution x are measured in the Frobenius norm. Numerical experiments that show the relevance of this condition number in the prediction of the computing error w...