Secure Beamforming for MIMO Two-Way Communications with an Untrusted Relay

This paper studies the secure beamforming design in a multiple-antenna three-node system where two source nodes exchange messages with the help of an untrusted relay node. The relay acts as both an essential signal forwarder and a potential eavesdropper. Both two-phase and three-phase two-way relay strategies are considered. Our goal is to jointly optimize the source and relay beamformers for maximizing the secrecy sum rate of the two-way communications. We first derive the optimal relay beamformer structures. Then, iterative algorithms are proposed to find source and relay beamformers jointly based on alternating optimization. Furthermore, we conduct asymptotic analysis on the maximum secrecy sum-rate. Our analysis shows that when all transmit powers approach infinity, the two-phase two-way relay scheme achieves the maximum secrecy sum rate if the source beamformers are designed such that the received signals at the relay align in the same direction. This reveals an important advantage of signal alignment technique in against eavesdropping. It is also shown that if the source powers approach zero the three-phase scheme performs the best while the two-phase scheme is even worse than direct transmission. Simulation results have verified the efficiency of the secure beamforming algorithms as well as the analytical findings.Comment: 10 figures, Submitted to IEEE Transactions on Signal Processin

Various identities for iterated integrals of a semimartingale

This paper derives several identities for iterated integrals of a semimartingale. They involve powers, brackets, exponential and the stochastic exponential. Some, like for counting or finite-variation processes, are apparently new. Others, like two of the three formulae for continuous semimartingales, are generalizations of well-known formulae

Brill-Gordan Loci, Transvectants and an Analogue of the Foulkes Conjecture

Combining a selection of tools from modern algebraic geometry, representation theory, the classical invariant theory of binary forms, together with explicit calculations with hypergeometric series and Feynman diagrams, we obtain the following interrelated results. A Castelnuovo-Mumford regularity bound and a projective normality result for the locus of hypersufaces that are equally supported on two hyperplanes. The surjectivity of an equivariant map between two plethystic compositions of symmetric powers; a statement which is reminiscent of the Foulkes-Howe conjecture. The nonvanishing of even transvectants of exact powers of generic binary forms. The nonvanishing of a collection of symmetric functions defined by sums over magic squares and transportation matrices with nonnegative integer entries. An explicit set of generators, in degree three, for the ideal of the coincident root locus of binary forms with only two roots of equal multiplicity.Comment: This is a considerably expanded version of math.AG/040523

Exact Finite-Size-Scaling Corrections to the Critical Two-Dimensional Ising Model on a Torus

We analyze the finite-size corrections to the energy and specific heat of the critical two-dimensional spin-1/2 Ising model on a torus. We extend the analysis of Ferdinand and Fisher to compute the correction of order L^{-3} to the energy and the corrections of order L^{-2} and L^{-3} to the specific heat. We also obtain general results on the form of the finite-size corrections to these quantities: only integer powers of L^{-1} occur, unmodified by logarithms (except of course for the leading $\log L$ term in the specific heat); and the energy expansion contains only odd powers of L^{-1}. In the specific-heat expansion any power of L^{-1} can appear, but the coefficients of the odd powers are proportional to the corresponding coefficients of the energy expansion.Comment: 26 pages (LaTeX). Self-unpacking file containing the tex file and three macros (indent.sty, eqsection.sty, subeqnarray.sty). Added discussions on the results and new references. Version to be published in J. Phys.

Relativistic corrections of one-nucleon current in low-energy three-nucleon photonuclear reactions

Proton-deuteron radiative capture and two- and three-body photodisintegration of 3He at low energy are described using realistic hadronic dynamics and including the Coulomb force. The sensitivity of the observables to the relativistic corrections of one-nucleon electromagnetic current operator is studied. Significant effects of the relativistic spin-orbit charge are found for the vector analyzing powers in the proton-deuteron radiative capture and for the beam-target parallel-antiparallel spin asymmetry in the three-body photodisintegration of 3He.Comment: 7 figures, accepted for publication in Phys. Rev.

On Waring's problem: two squares and three biquadrates

We investigate sums of mixed powers involving two squares and three biquadrates. In particular, subject to the truth of the Generalised Riemann Hypothesis and the Elliott-Halberstam Conjecture, we show that all large natural numbers n with 8 not dividing n, n not congruent to 2 modulo 3, and n not congruent to 14 modulo 16, are the sum of 2 squares and 3 biquadrates.Comment: to appear in Mathematik