A set K of nodes of a graph G is geodesically convex (respectively, monophonically convex) if K contains every node on every shortest (respectively, chordless) path joining nodes in K. We investigate the classes of graphs which are characterized by certain local convexity conditions with respect to geodesic convexity, in particular, those graphs in which balls around nodes are convex, and those...

Based on a classic paper by Ginibre [Commun. Math. Phys. 8, 26 (1968)] it is shown that, whenever Bogoliubov's approximation, that is, the replacement of a(0) and a*(0) by complex numbers in the Hamiltonian, asymptotically yields the right pressure, it also implies the asymptotic equality of |a(0)|(2)/V and a*(0)a(0)/V in symmetry breaking fields, irrespective of the existence or absence of Bos...

We study a sequential decision problem on a subset S ⊂ R. A decision maker chooses, on iteration t, a probability distribution π over S, then discovers a bounded loss function ` : S → [0,M ], and incurs the expectation Es∼π(t) `(s). The cumulative regret of the decision maker is then ∑T t=1 Es∼π(t) [`(x)] − infs∈S ∑t τ=1 ` (s). We investigate conditions under which one can guarantee a sublinear...

We study maximum penalized likelihood estimation for logistic regression type problems. The usual difficulties encountered when the log-odds ratios may become large in absolute value are circumvented by imposing a priori bounds on the estimator, depending on the sample size (n) and smoothing parameter. We pay for this in the convergence rate of the mean integrated squared error by a factor logn...

Abstract. We are interested in the Logarithmic Sobolev Inequality for the infinite volume Gibbs measure with no quadratic interactions. We consider unbounded spin systems on the one dimensional Lattice with interactions that go beyond the usual strict convexity and without uniform bound on the second derivative. We assume that the one dimensional single-site measure with boundaries satisfies th...

We describe an algorithm to find maximum-weight digitally convex regions in gray-scale images. After Christ et al. [CPS10] gave a system for consistent digital line systems, the question how to transfer the concept of convexity to digital regions naturally appeared. The regions computed using this definition are weight-maximal, the computational complexity of our algorithm amounts to O(n5 · log...