Melham's Conjecture on Odd Power Sums of Fibonacci Numbers

Ozeki and Prodinger showed that the odd power sum of the first several consecutive Fibonacci numbers of even order is equal to a polynomial evaluated at certain Fibonacci number of odd order. We prove that this polynomial and its derivative both vanish at $1$, and will be an integer polynomial after multiplying it by a product of the first consecutive Lucas numbers of odd order. This presents an affirmative answer to a conjecture of Melham.Comment: 15page

A major index for matchings and set partitions

We introduce a statistic \pmaj on partitions of $[n]=\{1,2,..., n\}$, and show that it is equidistributed with the number of 2-crossings over partitions of $[n]$ with given sets of minimal block elements and maximal block elements. This generalizes the classical result of equidistribution for the permutation statistics inversion number and major index.Comment: 17 pages, 9 figure

Multi-market minority game: breaking the symmetry of choice

Generalization of the minority game to more than one market is considered. At each time step every agent chooses one of its strategies and acts on the market related to this strategy. If the payoff function allows for strong fluctuation of utility then market occupancies become inhomogeneous with preference given to this market where the fluctuation occured first. There exists a critical size of agent population above which agents on bigger market behave collectively. In this regime there always exists a history of decisions for which all agents on a bigger market react identically.Comment: 15 pages, 12 figures, Accepted to 'Advances in Complex Systems

Heat Flow in the Arctic

Defines heat flow as the flux at the earth's solid surface of heat conducted from the interior; the heat-flow-unit (hfu) is on the order of 1-millionth calorie through each sq cm of the surface/sec, which is enough to melt a 4-mm layer of ice over the earth's surface/yr. Earth heat originates from radioactive decay of U, Th and K in the crust and mantle. Although land heat-flow measurements in the Arctic are too few for regional interpretation, those from Cape Thompson, Barrow and Cape Simpson, Northern Alaska are discussed and figured to show what they contribute to understanding of permafrost, climatic change and shoreline movements. Measuring thermal conductivity and gradient is much simpler in ocean basins than on land. Locations of such measurements are mapped, the results for the Alaskan quadrant in more detail. The sharp change in heat flow at the edge of the Alpha Cordillera, shown in a geothermal model, suggests that this feature is a huge accumulation of basalt, rather than mantle material or remnant of a foundering continent as previously postulated. Future Arctic heat flow studies are discussed

Your Survival Plan for Nuclear Fallout Emergency

Family and farm action plan worksheet for survival of a nuclear fallout emergency

Invariant tensors and Casimir operators for simple compact Lie groups

The Casimir operators of a Lie algebra are in one-to-one correspondence with the symmetric invariant tensors of the algebra. There is an infinite family of Casimir operators whose members are expressible in terms of a number of primitive Casimirs equal to the rank of the underlying group. A systematic derivation is presented of a complete set of identities expressing non-primitive symmetric tensors in terms of primitive tensors. Several examples are given including an application to an exceptional Lie algebra.Comment: 11 pages, LaTeX, minor changes, version in J. Math. Phy