A Meta-Algorithm for Creating Fast Algorithms for Counting ON Cells in Odd-Rule Cellular Automata

We develop a meta-algorithm that, given a polynomial (in one or more variables), and a prime p, produces a fast (logarithmic time) algorithm that takes a positive integer n and outputs the number of times each residue class modulo p appears as a coefficient when the polynomial is raised to the power n and the coefficients are read modulo p.Comment: 8 pages, accompanied by a Maple package, and numerous input and output files that can be gotten from http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/CAcount.htm

Universal quantum computation with unlabeled qubits

We show that an n-th root of the Walsh-Hadamard transform (obtained from the Hadamard gate and a cyclic permutation of the qubits), together with two diagonal matrices, namely a local qubit-flip (for a fixed but arbitrary qubit) and a non-local phase-flip (for a fixed but arbitrary coefficient), can do universal quantum computation on n qubits. A quantum computation, making use of n qubits and based on these operations, is then a word of variable length, but whose letters are always taken from an alphabet of cardinality three. Therefore, in contrast with other universal sets, no choice of qubit lines is needed for the application of the operations described here. A quantum algorithm based on this set can be interpreted as a discrete diffusion of a quantum particle on a de Bruijn graph, corrected on-the-fly by auxiliary modifications of the phases associated to the arcs.Comment: 6 page

Multiple planar coincidences with N-fold symmetry

Planar coincidence site lattices and modules with N-fold symmetry are well understood in a formulation based on cyclotomic fields, in particular for the class number one case, where they appear as certain principal ideals in the corresponding ring of integers. We extend this approach to multiple coincidences, which apply to triple or multiple junctions. In particular, we give explicit results for spectral, combinatorial and asymptotic properties in terms of Dirichlet series generating functions.Comment: 13 pages, two figures. For previous related work see math.MG/0511147 and math.CO/0301021. Minor changes and references update

On the Brownian gas: a field theory with a Poissonian ground state

As a first step towards a successful field theory of Brownian particles in interaction, we study exactly the non-interacting case, its combinatorics and its non-linear time-reversal symmetry. Even though the particles do not interact, the field theory contains an interaction term: the vertex is the hallmark of the original particle nature of the gas and it enforces the constraint of a strictly positive density field, as opposed to a Gaussian free field. We compute exactly all the n-point density correlation functions, determine non-perturbatively the Poissonian nature of the ground state and emphasize the futility of any coarse-graining assumption for the derivation of the field theory. We finally verify explicitly, on the n-point functions, the fluctuation-dissipation theorem implied by the time-reversal symmetry of the action.Comment: 31 page

Multiqubit symmetric states with high geometric entanglement

We propose a detailed study of the geometric entanglement properties of pure symmetric N-qubit states, focusing more particularly on the identification of symmetric states with a high geometric entanglement and how their entanglement behaves asymptotically for large N. We show that much higher geometric entanglement with improved asymptotical behavior can be obtained in comparison with the highly entangled balanced Dicke states studied previously. We also derive an upper bound for the geometric measure of entanglement of symmetric states. The connection with the quantumness of a state is discussed

A Fascinating Polynomial Sequence arising from an Electrostatics Problem on the Sphere

A positive unit point charge approaching from infinity a perfectly spherical isolated conductor carrying a total charge of +1 will eventually cause a negatively charged spherical cap to appear. The determination of the smallest distance $\rho(d)$ ($d$ is the dimension of the unit sphere) from the point charge to the sphere where still all of the sphere is positively charged is known as Gonchar's problem. Using classical potential theory for the harmonic case, we show that $1+\rho(d)$ is equal to the largest positive zero of a certain sequence of monic polynomials of degree $2d-1$ with integer coefficients which we call Gonchar polynomials. Rather surprisingly, $\rho(2)$ is the Golden ratio and $\rho(4)$ the lesser known Plastic number. But Gonchar polynomials have other interesting properties. We discuss their factorizations, investigate their zeros and present some challenging conjectures.Comment: 12 pages, 6 figures, 1 tabl

Masses of Astrometrically-Discovered and Imaged Binaries: G 78-28AB and GJ 231.1BC

The Stellar Planet Survey (STEPS) is an ongoing astrometric search for giant planets and brown dwarfs around a sample of ~30 M-dwarfs. We have discovered several low-mass companions by measuring the motion of our target stars relative to their reference frames. The highest mass discovery thus far is G 78-28B, a companion to the M-dwarf G 78-28A. The orbital period is 4.18 +/- 0.03 y, the system mass is 0.565 +/- 0.055 Msolar, and the semi-major axis is 2.19 +/- 0.10 AU. Imaging observations with the Keck laser guide star adaptive optics (LGSAO) and the Palomar AO instruments resolved the system and also yielded JHK-band delta magnitudes. We use the orbital solution, light ratios, and mass-luminosity relationships to derive component masses of MA = 0.370 +/- 0.034 Msolar and MB = 0.195 +/- 0.021 Msolar. G 78-28B is of type M4 V based upon its colors and mass. We also discovered GJ 231.1C, a companion to GJ 231.1B, with STEPS and imaged the companion with LGSAO and Palomar AO, but the orbital period is longer than our observing baseline; thus the system parameters are less constrained. In GJ 231.1BC the masses are MB = 0.25 +/- 0.06 Msolar and MC =0.12 +/- 0.02 Msolar. The inferred spectral type of GJ 231.1C is M5 V. We demonstrate the results of the current state of mass estimation techniques with our data.Comment: 25 pages, 8 figures, accepted for Ap

Revenue divergence and competitive balance in a divisional sports league

The North American model of resource allocation in professional sports leagues is adapted for English (association) football. The theoretical relationship between revenue and competitive balance is shown to be robust with respect to changes in teams’ objectives and labour market conditions. Empirical revenue functions are reported for 1926-1999. These indicate a shift in the composition of demand favouring big-city teams and an increase in the sensitivity of revenue to performance. An analysis of match results in the FA Cup competition suggests an increase in competitive imbalance between teams at different levels of the league’s divisional hierarchy, as the theory suggests