Burnside's lemma: A historical note

AbstractBurnside himself correctly ascribed the lemma now given his name to Frobenius. We explain how the subsequent error seems to have arisen

Creation and Growth of Components in a Random Hypergraph Process

Denote by an $\ell$-component a connected $b$-uniform hypergraph with $k$ edges and $k(b-1) - \ell$ vertices. We prove that the expected number of creations of $\ell$-component during a random hypergraph process tends to 1 as $\ell$ and $b$ tend to $\infty$ with the total number of vertices $n$ such that $\ell = o(\sqrt[3]{\frac{n}{b}})$. Under the same conditions, we also show that the expected number of vertices that ever belong to an $\ell$-component is approximately $12^{1/3} (b-1)^{1/3} \ell^{1/3} n^{2/3}$. As an immediate consequence, it follows that with high probability the largest $\ell$-component during the process is of size $O((b-1)^{1/3} \ell^{1/3} n^{2/3})$. Our results give insight about the size of giant components inside the phase transition of random hypergraphs.Comment: R\'{e}sum\'{e} \'{e}tend

Field Measurement of Light Penetration Through Sea Ice

In connection with phytoplankton studies, a non-optical, non-electric instrument has been devised for the measurement of relative light intensity in sea-ice bore holes. When used with a sensitive photometer, absolute values for the ambient light field can be determined within and immediately under the ice. As anticipated, attenuation is greatest at the ice-air interface; values just below the ice surface were 3 to 20% of incident. Another 70 to 100 cm of ice was required to effect a further 50% decrease in illumination. Extinction values were also measured on the ice cores in the laboratory, but scattering greatly complicates the interpretation of laboratory results

Testing Broken U(1) Symmetry in a Two-Component Atomic Bose-Einstein Condensate

We present a scheme for determining if the quantum state of a small trapped Bose-Einstein condensate is a state with well defined number of atoms, a Fock state, or a state with a broken U(1) gauge symmetry, a coherent state. The proposal is based on the observation of Ramsey fringes. The population difference observed in a Ramsey fringe experiment will exhibit collapse and revivals due to the mean-field interactions. The collapse and revival times depend on the relative strength of the mean-field interactions for the two components and the initial quantum state of the condensate.Comment: 20 Pages RevTex, 3 Figure

Solution of generalized fractional reaction-diffusion equations

This paper deals with the investigation of a closed form solution of a generalized fractional reaction-diffusion equation. The solution of the proposed problem is developed in a compact form in terms of the H-function by the application of direct and inverse Laplace and Fourier transforms. Fractional order moments and the asymptotic expansion of the solution are also obtained.Comment: LaTeX, 18 pages, corrected typo

Fractional reaction-diffusion equations

In a series of papers, Saxena, Mathai, and Haubold (2002, 2004a, 2004b) derived solutions of a number of fractional kinetic equations in terms of generalized Mittag-Leffler functions which provide the extension of the work of Haubold and Mathai (1995, 2000). The subject of the present paper is to investigate the solution of a fractional reaction-diffusion equation. The results derived are of general nature and include the results reported earlier by many authors, notably by Jespersen, Metzler, and Fogedby (1999) for anomalous diffusion and del-Castillo-Negrete, Carreras, and Lynch (2003) for reaction-diffusion systems with L\'evy flights. The solution has been developed in terms of the H-function in a compact form with the help of Laplace and Fourier transforms. Most of the results obtained are in a form suitable for numerical computation.Comment: LaTeX, 17 pages, corrected typo

Energies and collapse times of symmetric and symmetry-breaking states of finite systems with a U(1) symmetry

We study quantum systems of volume V, which will exhibit the breaking of a U(1) symmetry in the limit of V \to \infty, when V is large but finite. We estimate the energy difference between the `symmetric ground state' (SGS), which is the lowest-energy state that does not breaks the symmetry, and a `pure phase vacuum' (PPV), which approaches a symmetry-breaking vacuum as V \to \infty. Under some natural postulates on the energy of the SGS, it is shown that PPVs always have a higher energy than the SGS, and we derive a lower bound of the excess energy. We argue that the lower bound is O(V^0), which becomes much larger than the excitation energies of low-lying excited states for a large V. We also discuss the collapse time of PPVs for interacting many bosons. It is shown that the wave function collapses in a microscopic time scale, because PPVs are not energy eigenstates. We show, however, that for PPVs the expectation value of any observable, which is a finite polynomial of boson operators and their derivatives, does not collapse for a macroscopic time scale. In this sense, the collapse time of PPVs is macroscopically long.Comment: In the revised manuscript, Eq. (22), Ref. [8], and Notes [13], [15] and [17] have been adde

A Gaussian distribution for refined DT invariants and 3D partitions

We show that the refined Donaldson-Thomas invariants of C3, suitably normalized, have a Gaussian distribution as limit law. Combinatorially these numbers are given by weighted counts of 3D partitions. Our technique is to use the Hardy-Littlewood circle method to analyze the bivariate asymptotics of a q-deformation of MacMahon's function. The proof is based on that of E.M. Wright who explored the single variable case.Comment: 11 pages and 3 figure

Interference between the halves of a double-well trap containing a Bose-Einstein condensate

Interference between the halves of a double-well trap containing a Bose-Einstein condensate is studied. It is found that when the atoms in the two wells are initially in the coherent state, the intensity exhibits collapses and revivals, but it does not for the initial Fock states. Whether the initial states are in the coherent states or in a Fock states, the fidelity time has nothing to do with collision. We point out that interference and its fidelity can be adjusted experimentally by properly preparing the number and initial states of the system.Comment: 10 pages, 3 figures, accepted by Phy. rev.

Cooling the optical-spin driven limit cycle oscillations of a levitated gyroscope

Birefringent microspheres, trapped in vacuum and set into rotation by circularly polarised light, demonstrate remarkably stable translational motion. This is in marked contrast to isotropic particles in similar conditions. Here we demonstrate that this stability is obtained because the fast rotation of these birefringent spheres reduces the effect of azimuthal spin forces created by the inhomogeneous optical spin of circularly polarised light. At reduced pressures, the unique profile of these rotationally averaged, effective azimuthal forces results in the formation of nano-scale limit cycles. We demonstrate feedback cooling of these non-equilibrium oscillators, resulting in effective temperatures on the order of a milliKelvin. The principles we elaborate here can inform the design of high-stability rotors carrying enhanced centripetal loads or result in more efficient cooling schemes for autonomous limit cycle oscillations. Ultimately, this latter development could provide experimental access to non-equilibrium quantum effects within the mesoscopic regime.Yoshihiko Arita, Stephen H. Simpson, Graham D. Bruce, Ewan M. Wright, Pavel Zemánek, Kishan Dholaki