On the distribution of sums of residues

We generalize and solve the \roman{mod}\,q analogue of a problem of Littlewood and Offord, raised by Vaughan and Wooley, concerning the distribution of the $2^n$ sums of the form $\sum_{i=1}^n\varepsilon_ia_i$, where each $\varepsilon_i$ is $0$ or $1$. For all $q$, $n$, $k$ we determine the maximum, over all reduced residues $a_i$ and all sets $P$ consisting of $k$ arbitrary residues, of the number of these sums that belong to $P$.Comment: 5 page

Does the Gursey-Tze Solution Represent a Monopole Condensate?

We recast the quaternionic Gursey-Tze solution, which is a fourfold quasi-periodic self-dual Yang-Mills field with a unit instanton number per Euclidean spacetime cell, into an ordinary coordinate formulation. After performing the sum in the Euclidean time direction, we use an observation by Rossi which suggests the solution represents an arrangement with a BPS monopole per space lattice cell. This may provide a concrete realization of a monopole condensate in pure Yang-Mills theory.Comment: 12 pages, Latex

A new graph perspective on max-min fairness in Gaussian parallel channels

In this work we are concerned with the problem of achieving max-min fairness in Gaussian parallel channels with respect to a general performance function, including channel capacity or decoding reliability as special cases. As our central results, we characterize the laws which determine the value of the achievable max-min fair performance as a function of channel sharing policy and power allocation (to channels and users). In particular, we show that the max-min fair performance behaves as a specialized version of the Lovasz function, or Delsarte bound, of a certain graph induced by channel sharing combinatorics. We also prove that, in addition to such graph, merely a certain 2-norm distance dependent on the allowable power allocations and used performance functions, is sufficient for the characterization of max-min fair performance up to some candidate interval. Our results show also a specific role played by odd cycles in the graph induced by the channel sharing policy and we present an interesting relation between max-min fairness in parallel channels and optimal throughput in an associated interference channel.Comment: 41 pages, 8 figures. submitted to IEEE Transactions on Information Theory on August the 6th, 200

Effect of Dimensionality on the Percolation Thresholds of Various dd-Dimensional Lattices

We show analytically that the $[0,1]$, $[1,1]$ and $[2,1]$ Pad{\'e} approximants of the mean cluster number $S(p)$ for site and bond percolation on general $d$-dimensional lattices are upper bounds on this quantity in any Euclidean dimension $d$, where $p$ is the occupation probability. These results lead to certain lower bounds on the percolation threshold $p_c$ that become progressively tighter as $d$ increases and asymptotically exact as $d$ becomes large. These lower-bound estimates depend on the structure of the $d$-dimensional lattice and whether site or bond percolation is being considered. We obtain explicit bounds on $p_c$ for both site and bond percolation on five different lattices: $d$-dimensional generalizations of the simple-cubic, body-centered-cubic and face-centered-cubic Bravais lattices as well as the $d$-dimensional generalizations of the diamond and kagom{\'e} (or pyrochlore) non-Bravais lattices. These analytical estimates are used to assess available simulation results across dimensions (up through $d=13$ in some cases). It is noteworthy that the tightest lower bound provides reasonable estimates of $p_c$ in relatively low dimensions and becomes increasingly accurate as $d$ grows. We also derive high-dimensional asymptotic expansions for $p_c$ for the ten percolation problems and compare them to the Bethe-lattice approximation. Finally, we remark on the radius of convergence of the series expansion of $S$ in powers of $p$ as the dimension grows.Comment: 37 pages, 5 figure

Convex Clustering via Optimal Mass Transport

We consider approximating distributions within the framework of optimal mass transport and specialize to the problem of clustering data sets. Distances between distributions are measured in the Wasserstein metric. The main problem we consider is that of approximating sample distributions by ones with sparse support. This provides a new viewpoint to clustering. We propose different relaxations of a cardinality function which penalizes the size of the support set. We establish that a certain relaxation provides the tightest convex lower approximation to the cardinality penalty. We compare the performance of alternative relaxations on a numerical study on clustering.Comment: 12 pages, 12 figure