Exact Results on Potts Model Partition Functions in a Generalized External Field and Weighted-Set Graph Colorings

We present exact results on the partition function of the $q$-state Potts model on various families of graphs $G$ in a generalized external magnetic field that favors or disfavors spin values in a subset $I_s = \{1,...,s\}$ of the total set of possible spin values, $Z(G,q,s,v,w)$, where $v$ and $w$ are temperature- and field-dependent Boltzmann variables. We remark on differences in thermodynamic behavior between our model with a generalized external magnetic field and the Potts model with a conventional magnetic field that favors or disfavors a single spin value. Exact results are also given for the interesting special case of the zero-temperature Potts antiferromagnet, corresponding to a set-weighted chromatic polynomial $Ph(G,q,s,w)$ that counts the number of colorings of the vertices of $G$ subject to the condition that colors of adjacent vertices are different, with a weighting $w$ that favors or disfavors colors in the interval $I_s$. We derive powerful new upper and lower bounds on $Z(G,q,s,v,w)$ for the ferromagnetic case in terms of zero-field Potts partition functions with certain transformed arguments. We also prove general inequalities for $Z(G,q,s,v,w)$ on different families of tree graphs. As part of our analysis, we elucidate how the field-dependent Potts partition function and weighted-set chromatic polynomial distinguish, respectively, between Tutte-equivalent and chromatically equivalent pairs of graphs.Comment: 39 pages, 1 figur

Linear MMSE-Optimal Turbo Equalization Using Context Trees

Formulations of the turbo equalization approach to iterative equalization and decoding vary greatly when channel knowledge is either partially or completely unknown. Maximum aposteriori probability (MAP) and minimum mean square error (MMSE) approaches leverage channel knowledge to make explicit use of soft information (priors over the transmitted data bits) in a manner that is distinctly nonlinear, appearing either in a trellis formulation (MAP) or inside an inverted matrix (MMSE). To date, nearly all adaptive turbo equalization methods either estimate the channel or use a direct adaptation equalizer in which estimates of the transmitted data are formed from an expressly linear function of the received data and soft information, with this latter formulation being most common. We study a class of direct adaptation turbo equalizers that are both adaptive and nonlinear functions of the soft information from the decoder. We introduce piecewise linear models based on context trees that can adaptively approximate the nonlinear dependence of the equalizer on the soft information such that it can choose both the partition regions as well as the locally linear equalizer coefficients in each region independently, with computational complexity that remains of the order of a traditional direct adaptive linear equalizer. This approach is guaranteed to asymptotically achieve the performance of the best piecewise linear equalizer and we quantify the MSE performance of the resulting algorithm and the convergence of its MSE to that of the linear minimum MSE estimator as the depth of the context tree and the data length increase.Comment: Submitted to the IEEE Transactions on Signal Processin

Bar constructions for topological operads and the Goodwillie derivatives of the identity

We describe a cooperad structure on the simplicial bar construction on a reduced operad of based spaces or spectra and, dually, an operad structure on the cobar construction on a cooperad. We also show that if the homology of the original operad (respectively, cooperad) is Koszul, then the homology of the bar (respectively, cobar) construction is the Koszul dual. We use our results to construct an operad structure on the partition poset models for the Goodwillie derivatives of the identity functor on based spaces and show that this induces the `Lie' operad structure on the homology groups of these derivatives. We also extend the bar construction to modules over operads (and, dually, to comodules over cooperads) and show that a based space naturally gives rise to a left module over the operad formed by the derivatives of the identity.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper20.abs.html Version 3: Reference to Salvatore added (see footnote 3, page 834