Problems with Fitting to the Power-Law Distribution

This short communication uses a simple experiment to show that fitting to a power law distribution by using graphical methods based on linear fit on the log-log scale is biased and inaccurate. It shows that using maximum likelihood estimation (MLE) is far more robust. Finally, it presents a new table for performing the Kolmogorov-Smirnof test for goodness-of-fit tailored to power-law distributions in which the power-law exponent is estimated using MLE. The techniques presented here will advance the application of complex network theory by allowing reliable estimation of power-law models from data and further allowing quantitative assessment of goodness-of-fit of proposed power-law models to empirical data.Comment: 4 pages, 1 figure, 2 table

Symmetric M-ary phase discrimination using quantum-optical probe states

We present a theoretical study of minimum error probability discrimination, using quantum- optical probe states, of M optical phase shifts situated symmetrically on the unit circle. We assume ideal lossless conditions and full freedom for implementing quantum measurements and for probe state selection, subject only to a constraint on the average energy, i.e., photon number. In particular, the probe state is allowed to have any number of signal and ancillary modes, and to be pure or mixed. Our results are based on a simple criterion that partitions the set of pure probe states into equivalence classes with the same error probability performance. Under an energy constraint, we find the explicit form of the state that minimizes the error probability. This state is an unentangled but nonclassical single-mode state. The error performance of the optimal state is compared with several standard states in quantum optics. We also show that discrimination with zero error is possible only beyond a threshold energy of (M - 1)/2. For the M = 2 case, we show that the optimum performance is readily demonstrable with current technology. While transmission loss and detector inefficiencies lead to a nonzero erasure probability, the error rate conditional on no erasure is shown to remain the same as the optimal lossless error rate.Comment: 13 pages, 10 figure

The spontaneous emergence of ordered phases in crumpled sheets

X-ray tomography is performed to acquire 3D images of crumpled aluminum foils. We develop an algorithm to trace out the labyrinthian paths in the three perpendicular cross sections of the data matrices. The tangent-tangent correlation function along each path is found to decay exponentially with an effective persistence length that shortens as the crumpled ball becomes more compact. In the mean time, we observed ordered domains near the crust, similar to the lamellae phase mixed by the amorphous portion in lyotropic liquid crystals. The size and density of these domains grow with further compaction, and their orientation favors either perpendicular or parallel to the radial direction. Ordering is also identified near the core with an arbitrary orientation, exemplary of the spontaneous symmetry breaking

Extending Feynman's Formalisms for Modelling Human Joint Action Coordination

The recently developed Life-Space-Foam approach to goal-directed human action deals with individual actor dynamics. This paper applies the model to characterize the dynamics of co-action by two or more actors. This dynamics is modelled by: (i) a two-term joint action (including cognitive/motivatonal potential and kinetic energy), and (ii) its associated adaptive path integral, representing an infinite--dimensional neural network. Its feedback adaptation loop has been derived from Bernstein's concepts of sensory corrections loop in human motor control and Brooks' subsumption architectures in robotics. Potential applications of the proposed model in human--robot interaction research are discussed. Keywords: Psycho--physics, human joint action, path integralsComment: 6 pages, Late

Correlations in Bipartite Collaboration Networks

Collaboration networks are studied as an example of growing bipartite networks. These have been previously observed to have structure such as positive correlations between nearest-neighbour degrees. However, a detailed understanding of the origin of this phenomenon and the growth dynamics is lacking. Both of these are analyzed empirically and simulated using various models. A new one is presented, incorporating empirically necessary ingredients such as bipartiteness and sublinear preferential attachment. This, and a recently proposed model of team assembly both agree roughly with some empirical observations and fail in several others.Comment: 13 pages, 17 figures, 2 table, submitted to JSTAT; manuscript reorganized, figures and a table adde

Electron-lattice kinetics of metals heated by ultrashort laser pulses

We propose a kinetic model of transient nonequilibrium phenomena in metals exposed to ultrashort laser pulses when heated electrons affect the lattice through direct electron-phonon interaction. This model describes the destruction of a metal under intense laser pumping. We derive the system of equations for the metal, which consists of hot electrons and a cold lattice. Hot electrons are described with the help of the Boltzmann equation and equation of thermoconductivity. We use the equations of motion for lattice displacements with the electron force included. The lattice deformation is estimated immediately after the laser pulse up to the time of electron temperature relaxation. An estimate shows that the ablation regime can be achieved.Comment: 7 pages; Revtex. to appear in JETP 88, #1 (1999

Quantum SDP Solvers: Large Speed-ups, Optimality, and Applications to Quantum Learning

We give two quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with m constraint matrices, each of dimension n, rank at most r, and sparsity s. The first algorithm assumes access to an oracle to the matrices at unit cost. We show that it has run time Õ(s^2(√((mϵ)^(−10)) + √((nϵ)^(−12))), with ϵ the error of the solution. This gives an optimal dependence in terms of m, n and quadratic improvement over previous quantum algorithms when m ≈ n. The second algorithm assumes a fully quantum input model in which the matrices are given as quantum states. We show that its run time is Õ (√m + poly(r))⋅poly(log m,log n,B,ϵ^(−1)), with B an upper bound on the trace-norm of all input matrices. In particular the complexity depends only poly-logarithmically in n and polynomially in r. We apply the second SDP solver to learn a good description of a quantum state with respect to a set of measurements: Given m measurements and a supply of copies of an unknown state ρ with rank at most r, we show we can find in time √m⋅poly(log m,log n,r,ϵ^(−1)) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as ρ on the m measurements, up to error ϵ. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics. As in previous work, we obtain our algorithm by "quantizing" classical SDP solvers based on the matrix multiplicative weight method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians with a poly-logarithmic dependence on its dimension, which could be of independent interest

The Harmonic Measure for critical Potts clusters

We present a technique, which we call "etching," which we use to study the harmonic measure of Fortuin-Kasteleyn clusters in the Q-state Potts model for Q=1-4. The harmonic measure is the probability distribution of random walkers diffusing onto the perimeter of a cluster. We use etching to study regions of clusters which are extremely unlikely to be hit by random walkers, having hitting probabilities down to 10^(-4600). We find good agreement between the theoretical predictions of Duplantier and our numerical results for the generalized dimension D(q), including regions of small and negative q.Comment: 20 pages, 10 figure

Equation of state of resonance-rich matter in the central cell in heavy-ion collisions at s\sqrt{s}=200 AGeV

The equilibration of hot and dense nuclear matter produced in the central cell of central Au+Au collisions at RHIC ($\sqrt{s}=200$ AGeV) energies is studied within a microscopic transport model. The pressure in the cell becomes isotropic at $t\approx 5$ fm/$c$ after beginning of the collision. Within the next 15 fm/$c$ the expansion of matter in the cell proceeds almost isentropically with the entropy per baryon ratio $S/A \approx 150$, and the equation of state in the $(P,\epsilon)$ plane has a very simple form, $P=0.15\epsilon$. Comparison with the statistical model of an ideal hadron gas indicates that the time $t \approx 20$ fm/c may be too short to reach the fully equilibrated state. Particularly, the creation of long-lived resonance-rich matter in the cell decelerates the relaxation to chemical equilibrium. This resonance-abundant state can be detected experimentally after the thermal freeze-out of particles.Comment: LATEX, 21 pages incl. 7 figure