Achieving New Upper Bounds for the Hypergraph Duality Problem through Logic

The hypergraph duality problem DUAL is defined as follows: given two simple hypergraphs $\mathcal{G}$ and $\mathcal{H}$, decide whether $\mathcal{H}$ consists precisely of all minimal transversals of $\mathcal{G}$ (in which case we say that $\mathcal{G}$ is the dual of $\mathcal{H}$). This problem is equivalent to deciding whether two given non-redundant monotone DNFs are dual. It is known that non-DUAL, the complementary problem to DUAL, is in $\mathrm{GC}(\log^2 n,\mathrm{PTIME})$, where $\mathrm{GC}(f(n),\mathcal{C})$ denotes the complexity class of all problems that after a nondeterministic guess of $O(f(n))$ bits can be decided (checked) within complexity class $\mathcal{C}$. It was conjectured that non-DUAL is in $\mathrm{GC}(\log^2 n,\mathrm{LOGSPACE})$. In this paper we prove this conjecture and actually place the non-DUAL problem into the complexity class $\mathrm{GC}(\log^2 n,\mathrm{TC}^0)$ which is a subclass of $\mathrm{GC}(\log^2 n,\mathrm{LOGSPACE})$. We here refer to the logtime-uniform version of $\mathrm{TC}^0$, which corresponds to $\mathrm{FO(COUNT)}$, i.e., first order logic augmented by counting quantifiers. We achieve the latter bound in two steps. First, based on existing problem decomposition methods, we develop a new nondeterministic algorithm for non-DUAL that requires to guess $O(\log^2 n)$ bits. We then proceed by a logical analysis of this algorithm, allowing us to formulate its deterministic part in $\mathrm{FO(COUNT)}$. From this result, by the well known inclusion $\mathrm{TC}^0\subseteq\mathrm{LOGSPACE}$, it follows that DUAL belongs also to $\mathrm{DSPACE}[\log^2 n]$. Finally, by exploiting the principles on which the proposed nondeterministic algorithm is based, we devise a deterministic algorithm that, given two hypergraphs $\mathcal{G}$ and $\mathcal{H}$, computes in quadratic logspace a transversal of $\mathcal{G}$ missing in $\mathcal{H}$.Comment: Restructured the presentation in order to be the extended version of a paper that will shortly appear in SIAM Journal on Computin

External and internal influences on R&D alliance formation: evidence from German SMEs

Relying on relational capital theory and transaction cost economics (TCE), this study identifies factors that impede or promote alliance formation in small to medium-sized enterprises (SMEs). Environmental uncertainty and knowledge intensity impede firms’ R&D alliance formation; the focal firm’s overall trust in partners enhances alliance formation. Trust interacts positively with environmental uncertainty and knowledge intensity to affect alliance formation in SMEs. The findings reflect data from a longitudinal sample of 854 German SMEs, captured over eight years from 1999 to 2007

Neutrino Anomalies in an Extended Zee Model

We discuss an extended $SU(2)\times U(1)$ model which naturally leads to mass scales and mixing angles relevant for understanding both the solar and atmospheric neutrino anomalies. No right-handed neutrinos are introduced in the model.The model uses a softly broken $L_e-L_{\mu}-L_{\tau}$ symmetry. Neutrino masses arise only at the loop level. The one-loop neutrino masses which arise as in the Zee model solve the atmospheric neutrino anomaly while breaking of $L_e-L_{\mu}-L_{\tau}$ generates at two-loop order a mass splitting needed for the vacuum solution of the solar neutrino problem. A somewhat different model is possible which accommodates the large-angle MSW resolution of the solar neutrino problem.Comment: 11 pages including 2 figures; a reference added and text changed accordingl

Exact-diagonalization method for soft-core bosons in optical lattices using hierarchical wavefunctions

In this work, we describe a new technique for numerical exact-diagonalization. The method is particularly suitable for cold bosonic atoms in optical lattices, in which multiple atoms can occupy a lattice site. We describe the use of the method for Bose-Hubbard model as an example, however, the method is general and can be applied to other lattice models. The proposed numerical technique focuses in detail on how to construct the basis states as a hierarchy of wavefunctions. Starting from single-site Fock states we construct the basis set in terms of row-states and cluster-states. This simplifies the application of constraints and calculation of the Hamiltonian matrix. Each step of the method can be parallelized to accelerate the computation. In addition, we have illustrated the computation of the spatial bipartite entanglement entropy in the correlated $\nu =1/2$ fractional quantum Hall state.Comment: 11 pages, 6 figures, Comments are most welcom

A Shock Wave Attached to a Pointed Obstacle in a Steady flow of a Dissociating Gas

A pointed obstacle is assumed symmetrically placed with respect to a uniform supersonic flow ahead of it. It is assumed that an oblique shock wave attached to the leading edge of the obstacle appears form the vertex so that the flow after the shock is along the surface of the obstacle. In this paper a relation between the curvature of an attached shock wave and that of a stream line is discovered. It is concluded that in a steady flow of an ideal dissociating gas, the stream lines at the rear of a straight attached shock wave are necessarily curved lines, whereas in an ordinary gas flow only a straight line flow is possible behind a straight shock