On a Theorem of Sewell and Trotter

Sewell and Trotter [J. Combin. Theory Ser. B, 1993] proved that every connected alpha-critical graph that is not isomorphic to K_1, K_2 or an odd cycle contains a totally odd K_4-subdivision. Their theorem implies an interesting min-max relation for stable sets in graphs without totally odd K_4-subdivisions. In this note, we give a simpler proof of Sewell and Trotter's theorem.Comment: Referee comments incorporate

Remarks on the operator-norm convergence of the Trotter product formula

We revise the operator-norm convergence of the Trotter product formula for a pair {A,B} of generators of semigroups on a Banach space. Operator-norm convergence holds true if the dominating operator A generates a holomorphic contraction semigroup and B is a A-infinitesimally small generator of a contraction semigroup, in particular, if B is a bounded operator. Inspired by studies of evolution semigroups it is shown in the present paper that the operator-norm convergence generally fails even for bounded operators B if A is not a holomorphic generator. Moreover, it is shown that operator norm convergence of the Trotter product formula can be arbitrary slow.Comment: 12 page

Reductions for the Stable Set Problem

One approach to finding a maximum stable set (MSS) in a graph is to try to reduce the size of the problem by transforming the problem into an equivalent problem on a smaller graph. This paper introduces several new reductions for the MSS problem, extends several well-known reductions to the maximum weight stable set (MWSS) problem, demonstrates how reductions for the generalized stable set problem can be used in conjunction with probing to produce powerful new reductions for both the MSS and MWSS problems, and shows how hypergraphs can be used to expand the capabilities of clique projections. The effectiveness of these new reduction techniques are illustrated on the DIMACS benchmark graphs, planar graphs, and a set of challenging MSS problems arising from Steiner Triple Systems

Modular Groups of Quantum Fields in Thermal States

For a quantum field in a thermal equilibrium state we discuss the group generated by time translations and the modular action associated with an algebra invariant under half-sided translations. The modular flows associated with the algebras of the forward light cone and a space-like wedge admit a simple geometric description in two dimensional models that factorize in light-cone coordinates. At large distances from the domain boundary compared to the inverse temperature the flow pattern is essentially the same as time translations, whereas the zero temperature results are approximately reproduced close to the edge of the wedge and the apex of the cone. Associated with each domain there is also a one parameter group with a positive generator, for which the thermal state is a ground state. Formally, this may be regarded as a certain converse of the Unruh-effect.Comment: 28 pages, 4 figure

Persistency of Linear Programming Relaxations for the Stable Set Problem

The Nemhauser-Trotter theorem states that the standard linear programming (LP) formulation for the stable set problem has a remarkable property, also known as (weak) persistency: for every optimal LP solution that assigns integer values to some variables, there exists an optimal integer solution in which these variables retain the same values. While the standard LP is defined by only non-negativity and edge constraints, a variety of other LP formulations have been studied and one may wonder whether any of them has this property as well. We show that any other formulation that satisfies mild conditions cannot have the persistency property on all graphs, unless it is always equal to the stable set polytope.Comment: 17 pages, 6 figure

Enhancing the charging power of quantum batteries

Can collective quantum effects make a difference in a meaningful thermodynamic operation? Focusing on energy storage and batteries, we demonstrate that quantum mechanics can lead to an enhancement in the amount of work deposited per unit time, i.e., the charging power, when $N$ batteries are charged collectively. We first derive analytic upper bounds for the collective \emph{quantum advantage} in charging power for two choices of constraints on the charging Hamiltonian. We then highlight the importance of entanglement by proving that the quantum advantage vanishes when the collective state of the batteries is restricted to be in the separable ball. Finally, we provide an upper bound to the achievable quantum advantage when the interaction order is restricted, i.e., at most $k$ batteries are interacting. Our result is a fundamental limit on the advantage offered by quantum technologies over their classical counterparts as far as energy deposition is concerned.Comment: In this new updated version Theorem 1 has been changed with Proposition 1. The paper has been published on PRL, and DOI included accordingl