Let H be a complex Hilbert space and denote by Bs(H) the set of all self-adjoint bounded linear operators on H . In this paper we describe the form of all bijective maps (no linearity or continuity is assumed) on Bs(H) which preserve the order ≤ in both directions.

In most classification tasks there are observations that are ambiguous and therefore difficult to correctly label. Set-valued classification allows the classifiers to output a set of plausible labels rather than a single label, thereby giving a more appropriate and informative treatment to the labeling of ambiguous instances. We introduce a framework for multiclass set-valued classification, wh...

Given a graph G = (V, E) and a positive integer k, in MAXIMUM k-ORDER BOUNDED COMPONENT SET (MAX-k-OBCS), it is required to find a vertex set S ⊆ V of maximum size such that each component in the induced graph G[S] has at most k vertices. We prove that for constant k, MAX-k-OBCS is hard to approximate within a factor of n1−e, for any e > 0, unless P = NP. This is an improvement on the previous ...

This paper studies the problem of implementing a shared memory test-and-set object using only shared registers. Our contribution is threefold. First, we present a general framework to allow reasoning about reusing one-shot shared memory objects in the construction of bounded space long-lived objects. Then we derive general theorems about test-andset objects that simplifies reasoning about their...

Given an undirected planar graph G with n vertices and a set S of n points inside a simple polygon P , a point-set embedding of G on S is a planar drawing of G such that each vertex is mapped to a distinct point of S and the edges are polygonal chains surrounded by P . A special case of the embedding problem is that in which G is a balanced binary tree. In this paper, we present a new algorithm...

The principal term in the asymptotic expansion of the variance of the periodic measure of a ball in R under uniform random shift is proportional to the (d + 1)st power of the grid scaling factor. This result remains valid for a bounded set in R with sufficiently smooth isotropic covariogram under a uniform random shift and an isotropic rotation, and the asymptotic term is proportional also to t...

Let X be the minimal diffusion associated with a uniformly elliptic differential operator L on a bounded subdomain of Rd, with C2 boundary. Under the only assumption that the coefficients of L be Hölder continuous, we prove all the standard quasistationary limit theorems (cf. Markov chain theory). Moreover, we show that the laws of X, conditioned on explosion occuring after time s, converge in ...

We study approximation hardness of the Minimum Dominating Set problem and its variants in undirected and directed graphs. Using a similar result obtained by Trevisan for Minimum Set Cover we prove the first explicit approximation lower bounds for various kinds of domination problems (connected, total, independent) in bounded degree graphs. Asymptotically, for degree bound approaching infinity, ...

The Feedback Vertex Set problem (FVS), where the goal is to find a small subset of ver-tices that intersects every cycle in an input directed graph, is among the fundamental prob-lems whose approximability is not well-understood. One can efficiently find an Õ(log n)factor approximation, and while a constant-factor approximation is ruled out under theUnique Games Conjecture (UGC)...

Halldórsson (1995) claimed a (k + 1)/3 + 2 performance ratio for approximating the degree-bounded k-set packing problem. However, its detailed proof was omitted therein. In this report, we attempt to present in detail the proof toward this performance ratio which was briefly given in Halldórsson and Radhakrishnan (1994), and further show that the underlying local improvement search algorithm ru...