Finding the Minimum-Weight k-Path

Given a weighted $n$-vertex graph $G$ with integer edge-weights taken from a range $[-M,M]$, we show that the minimum-weight simple path visiting $k$ vertices can be found in time \tilde{O}(2^k \poly(k) M n^\omega) = O^*(2^k M). If the weights are reals in $[1,M]$, we provide a $(1+\varepsilon)$-approximation which has a running time of \tilde{O}(2^k \poly(k) n^\omega(\log\log M + 1/\varepsilon)). For the more general problem of $k$-tree, in which we wish to find a minimum-weight copy of a $k$-node tree $T$ in a given weighted graph $G$, under the same restrictions on edge weights respectively, we give an exact solution of running time \tilde{O}(2^k \poly(k) M n^3) and a $(1+\varepsilon)$-approximate solution of running time \tilde{O}(2^k \poly(k) n^3(\log\log M + 1/\varepsilon)). All of the above algorithms are randomized with a polynomially-small error probability.Comment: To appear at WADS 201

Spin-charge separation in ultra-cold quantum gases

We investigate the physical properties of quasi-1D quantum gases of fermion atoms confined in harmonic traps. Using the fact that for a homogeneous gas, the low energy properties are exactly described by a Luttinger model, we analyze the nature and manifestations of the spin-charge separation. Finally we discuss the necessary physical conditions and experimental limitations confronting possible experimental implementations.Comment: 4 pages, revtex4, 2 eps figure

Bounded Counter Languages

We show that deterministic finite automata equipped with $k$ two-way heads are equivalent to deterministic machines with a single two-way input head and $k-1$ linearly bounded counters if the accepted language is strictly bounded, i.e., a subset of $a_1^*a_2^*... a_m^*$ for a fixed sequence of symbols $a_1, a_2,..., a_m$. Then we investigate linear speed-up for counter machines. Lower and upper time bounds for concrete recognition problems are shown, implying that in general linear speed-up does not hold for counter machines. For bounded languages we develop a technique for speeding up computations by any constant factor at the expense of adding a fixed number of counters

NMR and Neutron Scattering Experiments on the Cuprate Superconductors: A Critical Re-Examination

We show that it is possible to reconcile NMR and neutron scattering experiments on both LSCO and YBCO, by making use of the Millis-Monien-Pines mean field phenomenological expression for the dynamic spin-spin response function, and reexamining the standard Shastry-Mila-Rice hyperfine Hamiltonian for NMR experiments. The recent neutron scattering results of Aeppli et al on LSCO (x=14%) are shown to agree quantitatively with the NMR measurements of $^{63}T_1$ and the magnetic scaling behavior proposed by Barzykin and Pines. The reconciliation of the $^{17}T_1$ relaxation rates with the degree of incommensuration in the spin fluctuation spectrum seen in neutron experiments is achieved by introducing a new transferred hyperfine coupling $C'$ between oxygen nuclei and their next nearest neighbor $Cu^{2+}$ spins; this leads to a near-perfect cancellation of the influence of the incommensurate spin fluctuation peaks on the oxygen relaxation rates of LSCO. The inclusion of the new $C'$ term also leads to a natural explanation, within the one-component model, the different temperature dependence of the anisotropic oxygen relaxation rates for different field orientations, recently observed by Martindale $et~al$. The measured significant decrease with doping of the anisotropy ratio, $R= ^{63}T_{1ab}/^{63}T_{1c}$ in LSCO system, from $R =3.9$ for ${\rm La_2CuO_4}$ to $R ~ 3.0$ for LSCO (x=15%) is made compatible with the doping dependence of the shift in the incommensurate spin fluctuation peaks measured in neutron experiments, by suitable choices of the direct and transferred hyperfine coupling constants $A_{\beta}$ and B.Comment: 24 pages in RevTex, 9 figures include

Charge transfer fluctuation, d−d-wave superconductivity, and the B1gB_{1g} Raman phonon in the Cuprates: A detailed analysis

The Raman spectrum of the $B_{1g}$ phonon in the superconducting cuprate materials is investigated theoretically in detail in both the normal and superconducting phases, and is contrasted with that of the $A_{1g}$ phonon. A mechanism involving the charge transfer fluctuation between the two oxygen ions in the CuO$_2$ plane coupled to the crystal field perpendicular to the plane is discussed and the resulting electron-phonon coupling is evaluated. Depending on the symmetry of the phonon the weight of different parts of the Fermi surface in the coupling is different. This provides the opportunity to obtain information on the superconducting gap function at certain parts of the Fermi surface. The lineshape of the phonon is then analyzed in detail both in the normal and superconducting states. The Fano lineshape is calculated in the normal state and the change of the linewidth with temperature below T$_{c}$ is investigated for a $d_{x^{2}-y^{2}}$ pairing symmetry. Excellent agreement is obtained for the $B_{1g}$ phonon lineshape in YBa$_{2}$Cu$_{3}$O$_{7}$. These experiments, however, can not distinguish between $d_{x^{2}-y^{2}}$ and a highly anisotropic $s$-wave pairing.Comment: Revtex, 21 pages + 4 postscript figures appended, tp

On rr-Simple kk-Path

An $r$-simple $k$-path is a {path} in the graph of length $k$ that passes through each vertex at most $r$ times. The $r$-SIMPLE $k$-PATH problem, given a graph $G$ as input, asks whether there exists an $r$-simple $k$-path in $G$. We first show that this problem is NP-Complete. We then show that there is a graph $G$ that contains an $r$-simple $k$-path and no simple path of length greater than $4\log k/\log r$. So this, in a sense, motivates this problem especially when one's goal is to find a short path that visits many vertices in the graph while bounding the number of visits at each vertex. We then give a randomized algorithm that runs in time $$\mathrm{poly}(n)\cdot 2^{O( k\cdot \log r/r)}$$ that solves the $r$-SIMPLE $k$-PATH on a graph with $n$ vertices with one-sided error. We also show that a randomized algorithm with running time $\mathrm{poly}(n)\cdot 2^{(c/2)k/ r}$ with $c<1$ gives a randomized algorithm with running time \poly(n)\cdot 2^{cn} for the Hamiltonian path problem in a directed graph - an outstanding open problem. So in a sense our algorithm is optimal up to an $O(\log r)$ factor

On the Bilayer Coupling in the Yttrium-Barium Family of High Temperature Superconductors

We present and solve a model for the susceptibility of two CuO2 planes coupled by an interplane coupling J_perp and use the results to analyze a recent "cross-relaxation" NMR experiment on Y2Ba4Cu7O15. We deduce that in this material the product of J_perp and the maximum value of the in-plane susceptibility chi_max varies from approximately 0.2 at T = 200 K to 0.4 at T = 120 K and that this implies the existence of a temperature dependent in-plane spin correlation length. Using estimates of chi_max from the literature we find 5 meV < J_perp < 20 meV. We discuss the relation of the NMR results to neutron scattering results which have been claimed to imply that in YBa2Cu3O_{6+x} the two planes of a bilayer are perfectly anticorrelated. We also propose that the recently observed 41 meV excitation in YBa2Cu3O7 is an exciton pulled down below the superconducting gap by J_perp.Comment: 11 pages, 3 postscript figures (uuencoded and compressed

Coexistence of Superconductivity and Antiferromagnetism in Multilayered High-TcT_c Superconductor HgBa2_2Ca4_4Cu5_5Oy_y: A Cu-NMR Study

We report a coexistence of superconductivity and antiferromagnetism in five-layered compound HgBa$_2$Ca$_4$Cu$_5$O$_y$ (Hg-1245) with $T_c=108$ K, which is composed of two types of CuO$_2$ planes in a unit cell; three inner planes (IP's) and two outer planes (OP's). The Cu-NMR study has revealed that the optimallydoped OP undergoes a superconducting (SC) transition at $T_c=108$ K, whereas the three underdoped IP's do an antiferromagnetic (AF) transition below $T_N\sim$ 60 K with the Cu moments of $\sim (0.3-0.4)\mu_B$. Thus bulk superconductivity with a high value of $T_c=108$ K and a static AF ordering at $T_N=60$ K are realized in the alternating AF and SC layers. The AF-spin polarization at the IP is found to induce the Cu moments of $\sim0.02\mu_B$ at the SC OP, which is the AF proximity effect into the SC OP.Comment: 6 pages, 8 figure

Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable

Recognition of minimal unsatisfiable CNF formulas (unsatisfiable CNF formulas which become satisfiable if any clause is removed) is a classical DP-complete problem. It was shown recently that minimal unsatisfiable formulas with n variables and n+k clauses can be recognized in time . We improve this result and present an algorithm with time complexity ; hence the problem turns out to be fixed-parameter tractable (FTP) in the sense of Downey and Fellows (Parameterized Complexity, 1999). Our algorithm gives rise to a fixed-parameter tractable parameterization of the satisfiability problem: If for a given set of clauses F, the number of clauses in each of its subsets exceeds the number of variables occurring in the subset at most by k, then we can decide in time whether F is satisfiable; k is called the maximum deficiency of F and can be efficiently computed by means of graph matching algorithms. Known parameters for fixed-parameter tractable satisfiability decision are tree-width or related to tree-width. Tree-width and maximum deficiency are incomparable in the sense that we can find formulas with constant maximum deficiency and arbitrarily high tree-width, and formulas where the converse prevails

First-Principles Calculations of Hyperfine Interactions in La_2CuO_4

We present the results of first-principles cluster calculations of the electronic structure of La_2CuO_4. Several clusters containing up to nine copper atoms embedded in a background potential were investigated. Spin-polarized calculations were performed both at the Hartree-Fock level and with density functional methods with generalized gradient corrections to the local density approximation. The distinct results for the electronic structure obtained with these two methods are discussed. The dependence of the electric-field gradients at the Cu and the O sites on the cluster size is studied and the results are compared to experiments. The magnetic hyperfine coupling parameters are carefully examined. Special attention is given to a quantitative determination of on-site and transferred hyperfine fields. We provide a detailed analysis that compares the hyperfine fields obtained for various cluster sizes with results from additional calculations of spin states with different multiplicities. From this we conclude that hyperfine couplings are mainly transferred from nearest neighbor Cu^{2+} ions and that contributions from further distant neighbors are marginal. The mechanisms giving rise to transfer of spin density are worked out. Assuming conventional values for the spin-orbit coupling, the total calculated hyperfine interaction parameters are compared to informations from experiments.Comment: 23 pages, 9 figure