Smale’s 17th problem asks: “Can a zero of
n
n
complex polynomial equations in
n
n
unknowns be found approximately, on the average, in polynomial time with a uniform algorithm?” We give a positive answer to this question. Namely, we describe a uniform probabilistic algorithm that computes an approximate zero of systems of polynomial equations
f
:
C
n
⟶
C
n
f:\mathbb {C}^n\longrightarrow \mathbb {C}^n
, performing a number of arithmetic operations which is polynomial in the size of the input, on the average.
We prove a new complexity bound, polynomial on the average, for the problem of finding an approximate zero of systems of polynomial equations. The average number of Newton steps required by this method is almost linear in the size of the input (dense encoding). We show that the method can also be used to approximate several or all the solutions of non-degenerate systems, and prove that this last task can be done in running time which is linear in the Bézout number of the system and polynomial in the size of the input, on the average.
Smale's 17th Problem asks "Can a zero of n complex polynomial equations in n unknowns be found approximately, on the average [for a suitable probability measure on the space of inputs], in polynomial time with a uniform algorithm?" We present a uniform probabilistic algorithm for this problem and prove that its complexity is polynomial. We thus obtain a partial positive solution to Smale 17th Problem.
We studied autistics by quantitative EEG spectral and coherence analysis during three experimental conditions: basal, watching a cartoon with audio (V–A), and with muted audio band (VwA). Significant reductions were found for the absolute power spectral density (PSD) in the central region for delta and theta, and in the posterior region for sigma and beta bands, lateralized to the right hemisphere. When comparing VwA versus the V–A in the midline regions, we found significant decrements of absolute PSD for delta, theta and alpha, and increments for the beta and gamma bands. In autistics, VwA versus V–A tended to show lower coherence values in the right hemisphere. An impairment of visual and auditory sensory integration in autistics might explain our results.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.