The polar decomposition of a square matrix has been generalized by several authors to scalar products on Rn or Cn given by a bilinear or sesquilinear form. Previous work has focused mainly on the case of square matrices, sometimes with the assumption of a Hermitian scalar product. We introduce the canonical generalized polar decomposition A = WS, defined for general m × n matrices A, where W is...

We report on the zonal variability of mesopelagic particulate organic carbon remineralization and deep carbon transfer potential during the Kerguelen Ocean and Plateau compared Study 2 expedition (KEOPS 2; October– November 2011) in an area of the polar front supporting recurrent massive blooms from natural Fe fertilization. Mesopelagic carbon remineralization (MR) was assessed using the excess...

We introduce and demonstrate a new mapping for generating music from the 2D Cellular Automata “Game of Life” (GL). The core of this mapping is based on a polar co-ordinate system with origin at the GL grid centre. Such a polar approach is designed to better capture the radial symmetry inherent in the rule set for GL. The radial symmetry of the GL rule set is key to the visual experience of obse...

A resolving set for a graph Γ is a collection of vertices S, chosen so that for each vertex v, the list of distances from v to the members of S uniquely specifies v. The metric dimension μ(Γ) is the smallest size of a resolving set for Γ. We consider the metric dimension of the dual polar graphs, and show that it is at most the rank over R of the incidence matrix of the corresponding polar spac...

This note shows that level sets of a function characterize an M-convex function introduced by Murota.

In this paper, a polar code for the m-user multiple access channel (MAC) with binary inputs is constructed. In particular, Arıkan’s polarization technique applied individually to each user will polarize any m-user binary input MAC into a finite collection of extremal MACs. The extremal MACs have a number of desirable properties: (i) the ‘uniform sum rate’ of the original channel is not lost, (i...

Polar codes are constructed for m-user multiple access channels (MAC) whose input alphabet size is a prime number. The block error probability under successive cancelation decoding is o(2−N 1/2− ), where N is the block length. Although the sum capacity is achieved by this coding scheme, some points in the symmetric capacity region may not be achieved. In the case where the channel is a combinat...

An eternal $m$-secure set of a graph $G = (V,E)$ is aset $S_0subseteq V$ that can defend against any sequence ofsingle-vertex attacks by means of multiple-guard shifts along theedges of $G$. A suitable placement of the guards is called aneternal $m$-secure set. The eternal $m$-security number$sigma_m(G)$ is the minimum cardinality among all eternal$m$-secure sets in $G$. An edge $uvin E(G)$ is ...

We propose the progressive rate-filling method as a framework to study agile construction of multilevel polar-coded modulation. show that bit indices within each component polar code can follow fixed, precomputed ranking sequence (e.g., Polar in 5G standard) while their allocated rates (i.e., number information bits code) be fast computed by exploiting target sum-rate approximation and proper m...