Lattice initial segments of the hyperdegrees

We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $\mathcal{D}_{h}$. In fact, we prove that every sublattice of any hyperarithmetic lattice (and so, in particular, every countable locally finite lattice) is isomorphic to an initial segment of $\mathcal{D}_{h}$. Corollaries include the decidability of the two quantifier theory of $$ and the undecidability of its three quantifier theory. The key tool in the proof is a new lattice representation theorem that provides a notion of forcing for which we can prove a version of the fusion lemma in the hyperarithmetic setting and so the preservation of $\omega _{1}^{CK}$. Somewhat surprisingly, the set theoretic analog of this forcing does not preserve $\omega _{1}$. On the other hand, we construct countable lattices that are not isomorphic to an initial segment of $\mathcal{D}_{h}$

The Ramsey Theory of Henson graphs

Analogues of Ramsey's Theorem for infinite structures such as the rationals or the Rado graph have been known for some time. In this context, one looks for optimal bounds, called degrees, for the number of colors in an isomorphic substructure rather than one color, as that is often impossible. Such theorems for Henson graphs however remained elusive, due to lack of techniques for handling forbidden cliques. Building on the author's recent result for the triangle-free Henson graph, we prove that for each $k\ge 4$, the $k$-clique-free Henson graph has finite big Ramsey degrees, the appropriate analogue of Ramsey's Theorem. We develop a method for coding copies of Henson graphs into a new class of trees, called strong coding trees, and prove Ramsey theorems for these trees which are applied to deduce finite big Ramsey degrees. The approach here provides a general methodology opening further study of big Ramsey degrees for ultrahomogeneous structures. The results have bearing on topological dynamics via work of Kechris, Pestov, and Todorcevic and of Zucker.Comment: 75 pages. Substantial revisions in the presentation. Submitte

Angle-dependence of quantum oscillations in YBa2Cu3O6.59 shows free spin behaviour of quasiparticles

Measurements of quantum oscillations in the cuprate superconductors afford a new opportunity to assess the extent to which the electronic properties of these materials yield to a description rooted in Fermi liquid theory. However, such an analysis is hampered by the small number of oscillatory periods observed. Here we employ a genetic algorithm to globally model the field, angular, and temperature dependence of the quantum oscillations observed in the resistivity of YBa2Cu3O6.59. This approach successfully fits an entire data set to a Fermi surface comprised of two small, quasi-2-dimensional cylinders. A key feature of the data is the first identification of the effect of Zeeman splitting, which separates spin-up and spin-down contributions, indicating that the quasiparticles in the cuprates behave as nearly free spins, constraining the source of the Fermi surface reconstruction to something other than a conventional spin density wave with moments parallel to the CuO2 planes.Comment: 8 pages, 4 figure

A Quantum Monte Carlo Method and Its Applications to Multi-Orbital Hubbard Models

We present a framework of an auxiliary field quantum Monte Carlo (QMC) method for multi-orbital Hubbard models. Our formulation can be applied to a Hamiltonian which includes terms for on-site Coulomb interaction for both intra- and inter-orbitals, intra-site exchange interaction and energy differences between orbitals. Based on our framework, we point out possible ways to investigate various phase transitions such as metal-insulator, magnetic and orbital order-disorder transitions without the minus sign problem. As an application, a two-band model is investigated by the projection QMC method and the ground state properties of this model are presented.Comment: 10 pages LaTeX including 2 PS figures, to appear in J.Phys.Soc.Jp

A method for the estimation of p-mode parameters from averaged solar oscillation power spectra

A new fitting methodology is presented which is equally well suited for the estimation of low-, medium-, and high-degree mode parameters from $m$-averaged solar oscillation power spectra of widely differing spectral resolution. This method, which we call the "Windowed, MuLTiple-Peak, averaged spectrum", or WMLTP Method, constructs a theoretical profile by convolving the weighted sum of the profiles of the modes appearing in the fitting box with the power spectrum of the window function of the observing run using weights from a leakage matrix that takes into account both observational and physical effects, such as the distortion of modes by solar latitudinal differential rotation. We demonstrate that the WMLTP Method makes substantial improvements in the inferences of the properties of the solar oscillations in comparison with a previous method that employed a single profile to represent each spectral peak. We also present an inversion for the internal solar structure which is based upon 6,366 modes that we have computed using the WMLTP method on the 66-day long 2010 SOHO/MDI Dynamics Run. To improve both the numerical stability and reliability of the inversion we developed a new procedure for the identification and correction of outliers in a frequency data set. We present evidence for a pronounced departure of the sound speed in the outer half of the solar convection zone and in the subsurface shear layer from the radial sound speed profile contained in Model~S of Christensen-Dalsgaard and his collaborators that existed in the rising phase of Solar Cycle~24 during mid-2010