A symmetric shortest-path table routing is a set of paths between all pairs of nodes in a graph such that the set is closed under path reversal, each path is a shortest path in the graph, and all paths with the same destination form a tree with a sink at the destination. Such a routing can be found for any finite, connected, undirected, positiveweighted graph by “salting” the edge weights so th...

We show that solving the frequency assignment problem is equivalent to solving a minimization problem involving spectral norms of matrices, if the link gain matrix is symmetric. Often the spectral norm is ideal for minimization problems, but not so in this case. We reformulate the minimization using the one-, innnity-and Frobenius norm. This allows us to deal with non symmetric link gain matric...

We consider the problem of constructing a cyclic listing of all bitstrings of length 2n+1 with Hamming weights in the interval [n+1−`, n+`], where 1 ≤ ` ≤ n+1, by flipping a single bit in each step. This is a far-ranging generalization of the wellknown middle two levels problem (the case ` = 1). We provide a solution for the case ` = 2 and solve a relaxed version of the problem for general valu...

Recently Laurie presented a new algorithm for the computation of (2n+1)-point Gauss-Kronrod quadrature rules with real nodes and positive weights. This algorithm first determines a symmetric tridiagonal matrix of order 2n+ 1 from certain mixed moments, and then computes a partial spectral factorization. We describe a new algorithm that does not require the entries of the tridiagonal matrix to b...

Abstract In this article, we present a survey of different types random walk models with local and non-local transitions on undirected weighted networks. We general approach by defining the dynamics as discrete-time Markovian process transition probabilities expressed in terms symmetric matrix weights. first part, describe matrices weights that define like normal walk, biased walks preferential...

We define and study sl2-categorifications on abelian categories. We show in particular that there is a self-derived (even homotopy) equivalence categorifying the adjoint action of the simple reflection. We construct categorifications for blocks of symmetric groups and deduce that two blocks are splendidly Rickard equivalent whenever they have isomorphic defect groups and we show that this impli...

Given a symmetric matrix what is the nearest correlation matrix, that is, the nearest symmetric positive semidefinite matrix with unit diagonal? This problem arises in the finance industry, where the correlations are between stocks. For distance measured in two weighted Frobenius norms we characterize the solution using convex analysis. We show how the modified alternating projections method ca...

Gradient backpropagation (BP) requires symmetric feedforward and feedback connections—the same weights must be used for forward and backward passes. This “weight transport problem” [1] is thought to be the crux of BP’s biological implausibility. Using 15 different classification datasets, we systematically study to what extent BP really depends on weight symmetry. Surprisingly, the results indi...