Root finding with threshold circuits

We show that for any constant d, complex roots of degree d univariate rational (or Gaussian rational) polynomials---given by a list of coefficients in binary---can be computed to a given accuracy by a uniform TC^0 algorithm (a uniform family of constant-depth polynomial-size threshold circuits). The basic idea is to compute the inverse function of the polynomial by a power series. We also discuss an application to the theory VTC^0 of bounded arithmetic.Comment: 19 pages, 1 figur

Real-time complexity constrained encoding

Complex software appliances can be deployed on hardware with limited available computational resources. This computational boundary puts an additional constraint on software applications. This can be an issue for real-time applications with a fixed time constraint such as low delay video encoding. In the context of High Efficiency Video Coding (HEVC), a limited number of publications have focused on controlling the complexity of an HEVC video encoder. In this paper, a technique is proposed to control complexity by deciding between 2Nx2N merge mode and full encoding, at different Coding Unit (CU) depths. The technique is demonstrated in two encoders. The results demonstrate fast convergence to a given complexity threshold, and a limited loss in rate-distortion performance (on average 2.84% Bjontegaard delta rate for 40% complexity reduction)

0-1 Integer Linear Programming with a Linear Number of Constraints

We give an exact algorithm for the 0-1 Integer Linear Programming problem with a linear number of constraints that improves over exhaustive search by an exponential factor. Specifically, our algorithm runs in time $2^{(1-\text{poly}(1/c))n}$ where n is the number of variables and cn is the number of constraints. The key idea for the algorithm is a reduction to the Vector Domination problem and a new algorithm for that subproblem

Neuronal circuitry for pain processing in the dorsal horn

Neurons in the spinal dorsal horn process sensory information, which is then transmitted to several brain regions, including those responsible for pain perception. The dorsal horn provides numerous potential targets for the development of novel analgesics and is thought to undergo changes that contribute to the exaggerated pain felt after nerve injury and inflammation. Despite its obvious importance, we still know little about the neuronal circuits that process sensory information, mainly because of the heterogeneity of the various neuronal components that make up these circuits. Recent studies have begun to shed light on the neuronal organization and circuitry of this complex region

Computers from plants we never made. Speculations

We discuss possible designs and prototypes of computing systems that could be based on morphological development of roots, interaction of roots, and analog electrical computation with plants, and plant-derived electronic components. In morphological plant processors data are represented by initial configuration of roots and configurations of sources of attractants and repellents; results of computation are represented by topology of the roots' network. Computation is implemented by the roots following gradients of attractants and repellents, as well as interacting with each other. Problems solvable by plant roots, in principle, include shortest-path, minimum spanning tree, Voronoi diagram, $\alpha$-shapes, convex subdivision of concave polygons. Electrical properties of plants can be modified by loading the plants with functional nanoparticles or coating parts of plants of conductive polymers. Thus, we are in position to make living variable resistors, capacitors, operational amplifiers, multipliers, potentiometers and fixed-function generators. The electrically modified plants can implement summation, integration with respect to time, inversion, multiplication, exponentiation, logarithm, division. Mathematical and engineering problems to be solved can be represented in plant root networks of resistive or reaction elements. Developments in plant-based computing architectures will trigger emergence of a unique community of biologists, electronic engineering and computer scientists working together to produce living electronic devices which future green computers will be made of.Comment: The chapter will be published in "Inspired by Nature. Computing inspired by physics, chemistry and biology. Essays presented to Julian Miller on the occasion of his 60th birthday", Editors: Susan Stepney and Andrew Adamatzky (Springer, 2017