Polyhedra with few 3-cuts are hamiltonian

In 1956, Tutte showed that every planar 4-connected graph is hamiltonian. In this article, we will generalize this result and prove that polyhedra with at most three 3-cuts are hamiltonian. In 2002 Jackson and Yu have shown this result for the subclass of triangulations. We also prove that polyhedra with at most four 3-cuts have a hamiltonian path. It is well known that for each $k \ge 6$ non-hamiltonian polyhedra with $k$ 3-cuts exist. We give computational results on lower bounds on the order of a possible non-hamiltonian polyhedron for the remaining open cases of polyhedra with four or five 3-cuts.Comment: 21 pages; changed titl

Opinion formation models on a gradient

Statistical physicists have become interested in models of collective social behavior such as opinion formation, where individuals change their inherently preferred opinion if their friends disagree. Real preferences often depend on regional cultural differences, which we model here as a spatial gradient $g$ in the initial opinion. The gradient does not only add reality to the model. It can also reveal that opinion clusters in two dimensions are typically in the standard (i.e.\ independent) percolation universality class, thus settling a recent controversy about a non-consensus model. However, using analytical and numerical tools, we also present a model where the width of the transition between opinions scales $\propto g^{-1/4}$, not $\propto g^{-4/7}$ as in independent percolation, and the cluster size distribution is consistent with first-order percolation.Comment: 12 pages, 8 figures, version accepted by PLoS ONE, online supplement added as appendi

Quantum error correction may delay, but also cause, entanglement sudden death

Dissipation may cause two initially entangled qubits to evolve into a separable state in a finite time. This behavior is called entanglement sudden death (ESD). We study to what extent quantum error correction can combat ESD. We find that in some cases quantum error correction can delay entanglement sudden death but in other cases quantum error correction may cause ESD for states that otherwise do not suffer from it. Our analysis also shows that fidelity may not be the best measure to compare the efficiency of different error correction codes since the fidelity is not directly coupled to a state's remaining entanglement.Comment: 3 figure

Transition from connected to fragmented vegetation across an environmental gradient: scaling laws in ecotone geometry

A change in the environmental conditions across space—for example, altitude or latitude—can cause significant changes in the density of a vegetation type and, consequently, in spatial connectivity. We use spatially explicit simulations to study the transition from connected to fragmented vegetation. A static (gradient percolation) model is compared to dynamic (gradient contact process) models. Connectivity is characterized from the perspective of various species that use this vegetation type for habitat and differ in dispersal or migration range, that is, “step length” across the landscape. The boundary of connected vegetation delineated by a particular step length is termed the “ hull edge.” We found that for every step length and for every gradient, the hull edge is a fractal with dimension 7/4. The result is the same for different spatial models, suggesting that there are universal laws in ecotone geometry. To demonstrate that the model is applicable to real data, a hull edge of fractal dimension 7/4 is shown on a satellite image of a piñon‐juniper woodland on a hillside. We propose to use the hull edge to define the boundary of a vegetation type unambiguously. This offers a new tool for detecting a shift of the boundary due to a climate change

Entanglement invariant for the double Jaynes-Cummings model

We study entanglement dynamics between four qubits interacting through two isolated Jaynes-Cummings hamiltonians, via the entanglement measure based on the wedge product. We compare the results with similar results obtained using bipartite concurrence resulting in what is referred to as "entanglement sudden death". We find a natural entanglement invariant under evolution demonstrating that entanglement sudden death is caused by ignoring (tracing over) some of the system's degrees of freedom that become entangled through the interaction.Comment: Sec. V has largely been rewritten. An error pertaining to the entanglement invariant has been corrected and a correct invariant valid for a much larger set of states have been found, Eq. (25

Composite Fermions in Negative Effective Magnetic Field: A Monte-Carlo Study

The method of Jain and Kamilla [PRB {\bf 55}, R4895 (1997)] allows numerical generation of composite fermion trial wavefunctions for large numbers of electrons in high magnetic fields at filling fractions of the form nu=p/(2mp+1) with m and p positive integers. In the current paper we generalize this method to the case where the composite fermions are in an effective (mean) field with opposite sign from the actual physical field, i.e. when p is negative. We examine both the ground state energies and the low energy neutral excitation spectra of these states. Using particle-hole symmetry we can confirm the correctness of our method by comparing results for the series m=1 with p>0 (previously calculated by others) to our results for the conjugate series m=1 with p <0. Finally, we present similar results for ground state energies and low energy neutral excitations for the states with m=2 and p <0 which were not previously addressable, comparing our results to the m=1 case and the p > 0, m=2 cases.Comment: 11 page

The pH-dependent tertiary structure of a designed helix–loop–helix dimer

Background: De novo designed helix–loop–helix motifs can fold into well-defined tertiary structures if residues or groups of residues are incorporated at the helix–helix boundary to form helix-recognition sites that restrict the conformational degrees of freedom of the helical segments. Understanding the relationship between structure and function of conformational constraints therefore forms the basis for the engineering of non-natural proteins. This paper describes the design of an interhelical HisH+–Asp- hydrogen-bonded ion pair and the conformational stability of the folded helix–loop–helix motif.Results: GTD-C, a polypeptide with 43 amino acid residues, has been designed to fold into a hairpin helix–loop–helix motif that can dimerise to form a four-helix bundle. The folded motif is in slow conformational exchange on the NMR timescale and has a well-dispersed 1H NMR spectrum, a narrow temperature interval for thermal denaturation and a near-UV CD spectrum with some fine structure. The conformational stability is pH dependent with an optimum that corresponds to the pH for maximum formation of a hydrogen-bonded ion pair between HisH17+ in helix I and Asp27- in helix II.Conclusions: The formation of an interhelical salt bridge is strongly suggested by the pH dependence of a number of spectroscopic probes to generate a well-defined tertiary structure in a designed helix–loop–helix motif. The thermodynamic stability of the folded motif is not increased by the formation of the salt bridge, but neighbouring conformations are destabilised. The use of this novel design principle in combination with hydrophobic interactions that provide sufficient binding energy in the folded structure should be of general use in de novo design of native-like proteins