We deterministically construct quasi-polynomial weights in quasi-polynomial time, such that in a given polytope with totally unimodular constraints, one vertex is isolated, i.e., there is a unique minimum weight vertex. More precisely, the property that we need is that every face of the polytope lies in an affine space defined by a totally unimodular matrix. This derandomizes the famous Isolati...

Classical inequalities used in information theory such as those of de Bruijn, Fisher, Cramér, Rao, and Kullback carry over in a natural way from Euclidean space to unimodular Lie groups. These are groups that possess an integration measure that is simultaneously invariant under left and right shifts. All commutative groups are unimodular. And even in noncommutative cases unimodular Lie groups s...

We consider mixed-integer sets of the type MIX = {x : Ax ≥ b; xi integer, i ∈ I}, where A is a totally unimodular matrix, b is an arbitrary vector and I is a nonempty subset of the column indices of A. We show that the problem of checking nonemptiness of a setMIX is NP-complete even in the case in which the system describes mixed-integer network flows with half-integral requirements on the node...

We propose a unimodular version of the Brans–Dicke theory designed with constrained Lagrangian formulation. The resulting field equations are traceless. vacuum solutions in cosmological background reproduce corresponding usual but constant term. A perturbative analysis scalar modes is performed and stable unstable configurations appear, contrast case for which only occur. On other hand, tensori...

We show that for infinite planar unimodular random rooted maps, many global geometric and probabilistic properties are equivalent, and are determined by a natural, local notion of average curvature. This dichotomy includes properties relating to amenability, conformal geometry, random walks, uniform and minimal spanning forests, and Bernoulli bond percolation. We also prove that every simply co...

In this work it is shown that a necessary condition for the completeness of the geodesics of left invariant pseudo-Riemannian metrics on Lie groups is also sufficient in the case of 3-dimensional unimodular Lie groups, and not sufficient for 3-dimensional non unimodular Lie groups. As a consequence it is possible to identify, amongst the compact locally homogeneous Lorentzian 3-manifolds with n...

We define binet matrices, which furnish a direct generalization of totally unimodular network matrices and arise from the node-edge incidence matrices of bidirected graphs in the same way as the network matrices do from directed graphs. We develop the necessary theory, give binet representations for interesting sets of matrices, characterize totally unimodular binet matrices and discuss the rec...

In this paper, we propose the palindromic doubling algorithm (PDA) for the palindromic generalized eigenvalue problem (PGEP) A∗x = λAx. We establish a complete convergence theory of the PDA for PGEPs without unimodular eigenvalues, or with unimodular eigenvalues of partial multiplicities two (one or two for eigenvalue 1). Some important applications from the vibration analysis and the optimal c...