Similarity classes of 3x3 matrices over a local principal ideal ring

In this paper similarity classes of three by three matrices over a local principal ideal commutative ring are analyzed. When the residue field is finite, a generating function for the number of similarity classes for all finite quotients of the ring is computed explicitly.Comment: 14 pages, final version, to appear in Communications in Algebr

LU factorizations, q=0 limits, and p-adic interpretations of some q-hypergeometric orthogonal polynomials

For little q-Jacobi polynomials and q-Hahn polynomials we give particular q-hypergeometric series representations in which the termwise q=0 limit can be taken. When rewritten in matrix form, these series representations can be viewed as LU factorizations. We develop a general theory of LU factorizations related to complete systems of orthogonal polynomials with discrete orthogonality relations which admit a dual system of orthogonal polynomials. For the q=0 orthogonal limit functions we discuss interpretations on p-adic spaces. In the little 0-Jacobi case we also discuss product formulas.Comment: changed title, references updated, minor changes matching the version to appear in Ramanujan J.; 22 p

Nonlinear Matroid Optimization and Experimental Design

We study the problem of optimizing nonlinear objective functions over matroids presented by oracles or explicitly. Such functions can be interpreted as the balancing of multi-criteria optimization. We provide a combinatorial polynomial time algorithm for arbitrary oracle-presented matroids, that makes repeated use of matroid intersection, and an algebraic algorithm for vectorial matroids. Our work is partly motivated by applications to minimum-aberration model-fitting in experimental design in statistics, which we discuss and demonstrate in detail

A polynomial-time algorithm for optimizing over N-fold 4-block decomposable integer programs

In this paper we generalize N-fold integer programs and two-stage integer programs with N scenarios to N-fold 4-block decomposable integer programs. We show that for fixed blocks but variable N, these integer programs are polynomial-time solvable for any linear objective. Moreover, we present a polynomial-time computable optimality certificate for the case of fixed blocks, variable N and any convex separable objective function. We conclude with two sample applications, stochastic integer programs with second-order dominance constraints and stochastic integer multi-commodity flows, which (for fixed blocks) can be solved in polynomial time in the number of scenarios and commodities and in the binary encoding length of the input data. In the proof of our main theorem we combine several non-trivial constructions from the theory of Graver bases. We are confident that our approach paves the way for further extensions

A polynomial oracle-time algorithm for convex integer minimization

In this paper we consider the solution of certain convex integer minimization problems via greedy augmentation procedures. We show that a greedy augmentation procedure that employs only directions from certain Graver bases needs only polynomially many augmentation steps to solve the given problem. We extend these results to convex $N$-fold integer minimization problems and to convex 2-stage stochastic integer minimization problems. Finally, we present some applications of convex $N$-fold integer minimization problems for which our approach provides polynomial time solution algorithms.Comment: 19 pages, 1 figur

Low temperature spin fluctuations in geometrically frustrated Yb3Ga5O12

In the garnet structure compound Yb3Ga5O12, the Yb3+ ions (ground state effective spin S' = 1/2) are situated on two interpenetrating corner sharing triangular sublattices such that frustrated magnetic interactions are possible. Previous specific heat measurements evidenced the development of short range magnetic correlations below 0.5K and a lambda-transition at 54mK (Filippi et al. J. Phys. C: Solid State Physics 13 (1980) 1277). From 170-Yb M"ossbauer spectroscopy measurements down to 36mK, we find there is no static magnetic order at temperatures below that of the lambda-transition. Below 0.3K, the fluctuation frequency of the short range correlated Yb3+ moments progressively slows down and as the temperature tends to 0, the frequency tends to a quasi-saturated value of 3 x 10^9 s^-1. We also examined the Yb3+ paramagnetic relaxation rates up to 300K using 172-Yb perturbed angular correlation measurements: they evidence phonon driven processes.Comment: 6 pages, 5 figure

Learning intrinsic excitability in medium spiny neurons

We present an unsupervised, local activation-dependent learning rule for intrinsic plasticity (IP) which affects the composition of ion channel conductances for single neurons in a use-dependent way. We use a single-compartment conductance-based model for medium spiny striatal neurons in order to show the effects of parametrization of individual ion channels on the neuronal activation function. We show that parameter changes within the physiological ranges are sufficient to create an ensemble of neurons with significantly different activation functions. We emphasize that the effects of intrinsic neuronal variability on spiking behavior require a distributed mode of synaptic input and can be eliminated by strongly correlated input. We show how variability and adaptivity in ion channel conductances can be utilized to store patterns without an additional contribution by synaptic plasticity (SP). The adaptation of the spike response may result in either "positive" or "negative" pattern learning. However, read-out of stored information depends on a distributed pattern of synaptic activity to let intrinsic variability determine spike response. We briefly discuss the implications of this conditional memory on learning and addiction.Comment: 20 pages, 8 figure

Small grid embeddings of 3-polytopes

We introduce an algorithm that embeds a given 3-connected planar graph as a convex 3-polytope with integer coordinates. The size of the coordinates is bounded by $O(2^{7.55n})=O(188^{n})$. If the graph contains a triangle we can bound the integer coordinates by $O(2^{4.82n})$. If the graph contains a quadrilateral we can bound the integer coordinates by $O(2^{5.46n})$. The crucial part of the algorithm is to find a convex plane embedding whose edges can be weighted such that the sum of the weighted edges, seen as vectors, cancel at every point. It is well known that this can be guaranteed for the interior vertices by applying a technique of Tutte. We show how to extend Tutte's ideas to construct a plane embedding where the weighted vector sums cancel also on the vertices of the boundary face

Thermodynamic Study of Excitations in a 3D Spin Liquid

In order to characterize thermal excitations in a frustrated spin liquid, we have examined the magnetothermodynamics of a model geometrically frustrated magnet. Our data demonstrate a crossover in the nature of the spin excitations between the spin liquid phase and the high-temperature paramagnetic state. The temperature dependence of both the specific heat and magnetization in the spin liquid phase can be fit within a simple model which assumes that the spin excitations have a gapped quadratic dispersion relation.Comment: 5 figure