arXiv subjects
Publications and source records attributed to Paul Larson.
We produce a model of ZF + DC in which there exists a discontinuous homomorphism from the real line to itself but no Hamel basis for the real line, and prove a generalization of this result in terms of internal direct sums.
We force the Axiom of Choice over the least initial segment of a Nairian model satisfying ZF. In the forcing extension, square_kappa fails at all uncountable cardinals kappa, and every regular cardinal is omega-strongly measurable in HOD, as witnessed by the omega-club filter. Thus the failure of square everywhere is within the current reach of inner model theory, and the HOD Hypothesis is not provable in ZFC.
Using cloud Database as a Service (DBaaS) offerings instead of on-premise deployments is increasingly common. Key advantages include improved availability and scalability at a lower cost than on-premise alternatives. In this paper, we describe the design of Taurus, a new multi-tenant cloud database system. Taurus separates the compute and storage layers in a similar manner to Amazon Aurora and Microsoft Socrates and provides similar benefits, such as read replica support, low network utilization, hardware sharing and scalability. However, the Taurus architecture has several unique advantages. Taurus offers novel replication and recovery algorithms providing better availability than existing approaches using the same or fewer replicas. Also, Taurus is highly optimized for performance, using no more than one network hop on critical paths and exclusively using append-only storage, delivering faster writes, reduced device wear, and constant-time snapshots. This paper describes Taurus and provides a detailed description and analysis of the storage node architecture, which has not been previously available from the published literature.
We produce, relative to a ${\sf ZFC}$ model with a supercompact cardinal, a ${\sf ZFC}$ model of the Proper Forcing Axiom in which the nonstationary ideal on $\omega_1$ is $\Pi_1$-definable in a parameter from $H_{\aleph_2}$.
We prove a strong dichotomy result for countably-infinite oriented graphs; that is, we prove that for all countably-infinite oriented graphs $G$, either (i) there is a countably-infinite tournament $K$ such that $G\not\subseteq K$, or (ii) every countably-infinite tournament contains a \emph{spanning} copy of $G$. Furthermore, we are able to give a concise characterization of such oriented graphs. Our characterization becomes even simpler in the case of transitive acyclic oriented graphs (i.e. strict partial orders). For uncountable oriented graphs, we are able to extend the dichotomy result mentioned above to all regular cardinals $\kappa$; however, we are only able to provide a concise characterization in the case when $\kappa=\aleph_1$.
We show that under $\BMM$ and "there exists a Woodin cardinal$"$, the nonstationary ideal on $\omega_1$ can not be defined by a $\Sigma_1$ formula with parameter $A \subset \omega_1$. We show that the same conclusion holds under the assumption of Woodin's $(\ast)$-axiom. We further show that there are universes where $\BPFA$ holds and $\NS$ is $\Sigma_1(\omega_1)$-definable. Last we show that if the canonical inner model with one Woodin cardinal $M_1$ exists, there is a universe where $\NS$ is saturated, $\Sigma_1(\omega_1)$-definable and $\MA$ holds.
The high-pressure properties of fluorine and chlorine are not yet well understood because both are highly reactive and volatile elements, which has made conducting diamond anvil cell and x-ray diffraction experiments a challenge. Here we use ab initio methods to search for stable crystal structures of both elements at megabar pressures. We demonstrate how symmetry and geometric constraints can be combined to efficiently generate crystal structures that are composed of diatomic molecules. Our algorithm extends the symmetry driven structure search method [Phys. Rev. B 98 (2018) 174107] by adding constraints for the bond length and the number of atoms in a molecule, while still maintaining generality. As a method of validation, we have tested our approach for dense hydrogen and reproduced the known molecular structures of Cmca-12 and Cmca-4. We apply our algorithm to study chlorine and fluorine in the pressure range from 10--4000 GPa while considering crystal structures with up to 40 atoms per unit cell. We predict chlorine to follow the same series of phase transformations as elemental iodine from Cmca to Immm to Fm$\bar{3}$m, but at substantially higher pressures. We predict fluorine to transition from a C2/c to an Cmca structure at 70 GPa, to a novel orthorhombic and metallic structure with P$4_2$/mmc symmetry at 2500 GPa, and finally into its cubic analogue form with Pm$\bar{3}$n symmetry at 3000 GPa.
We show that if $\gamma \in \omega \cup \{\aleph_{0}\}$ and $\mathcal{A}$ is a counterexample to the analytic Vaught conjecture having exactly $\gamma$ many models of Scott rank $\omega_{1}$, then there exists a club $C \subseteq \omega_{1}$ such that $\mathcal{A}$ has exactly $\gamma$ many models of Scott rank $\alpha$, for each $\alpha \in C$.
We study conditions on automorphisms of Boolean algebras of the form $P(\lambda)/I_\kappa$ (where $\lambda$ is an uncountable cardinal and $I_\kappa$ is the ideal of sets of cardinality less than $\kappa$) which allow one to conclude that a given automorphism is trivial. We show (among other things) that every cardinality-preserving automorphism of $P(2^\kappa)/I_{\kappa^+}$ which is trivial on all sets of cardinality $\kappa^+$ is trivial, and that $MA_{\aleph_1}$ implies that every automorphism of $P(\mathbb{R})/Fin$ is trivial on a cocountable set.
By forcing with $\mathbb{P}_{\rm max}$ over strong models of determinacy, we obtain models where different square principles at $\omega_2$ and $\omega_3$ fail. In particular, we obtain a model of $2^{\aleph_0}=2^{\aleph_1}=\aleph_2 + \lnot\square(\omega_2) + \lnot\square(\omega_3)$.
We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space which does not include a perfect pre-image of omega_1 is hereditarily paracompact.
We show that that a certain class of semi-proper iterations does not add omega-sequences. As a result, starting from suitable large cardinals one can obtain a model in which the Continuum Hypothesis holds and every function from omega_1 to omega_1 is bounded by a canonical function on a club, and so omega_1 is the omega_2-nd canonical function.
Suppose we are given a family of choice functions on pairs from a given finite set. The set is considered as a set of alternatives (say candidates for an office) and the functions as potential "voters". The question is, what choice functions agree, on every pair, with the majority of some finite subfamily of the voters? For the problem as stated, a complete characterization was given in \citet{shelah2009mdp}, but here we allow each voter to abstain. There are four cases.
Working in the context of restricted forms of the Axiom of Choice, we consider the problem of splitting the ordinals below $λ$ of cofinality $θ$ into $λ$ many stationary sets, where $θ< λ$ are regular cardinals. This is a continuation of \cite{Sh835}.
A subset of a topological space is said to be \emph{universally measurable} if it is measured by the completion of each countably additive $σ$-finite Borel measure on the space, and \emph{universally null} if it has measure zero for each such atomless measure. In 1908, Hausdorff proved that there exist universally null sets of real numbers of cardinality $\aleph_{1}$, and thus that there exist at least $2^{\aleph_{1}}$ such sets. Laver showed in the 1970's that consistently there are just continuum many universally null sets of reals. The question of whether there exist more than continuum many universally measurable sets of reals was asked by Mauldin in 1978. We show that consistently there exist only continuum many universally measurable sets. This result also follows from work of Ciesielski and Pawlikowski on the iterated Sacks model. In the models we consider (forcing extensions by suitably-sized random algebras) every set of reals is universally measurable if and only if it and its complement are unions of ground model continuum many Borel sets.
The \emph{stationary set splitting game} is a game of perfect information of length $ω_{1}$ between two players, \unspls and \spl, in which \unspls chooses stationarily many countable ordinals and \spls tries to continuously divide them into two stationary pieces. We show that it is possible in ZFC to force a winning strategy for either player, or for neither. This gives a new counterexample to $Σ^{2}_{2}$ maximality with a predicate for the nonstationary ideal on $ω_{1}$, and an example of a consistently undetermined game of length $ω_{1}$ with payoff definable in the second-order monadic logic of order. We also show that the determinacy of the game is consistent with Martin's Axiom but not Martin's Maximum.
A computational study of the electronic structure and magnetic properties of Gd-pnictides is reported. The calculations were performed using a full-potential linear muffin-tin orbital (LMTO) method within the so-called LSDA+U approach, which adds Hubbard-U correlation effects to specified narrow bands in a mean-field approach to the local spin density approximation (LSDA). Here both the Gd 4f and 5d states are subject to such corrections. The U(f) values were determined semi-empirically by using photo-emission and inverse photoemission data for GdP, GdAs, GdSb and GdBi. In contrast to U(f) which represents narrow band physics, U(d) represents a quasiparticle self-energy correction of the LSDA gap underestimate. The U(d) value was adjusted using optical absorption data for semiconducting GdN above its Curie temperature. Below the Curie temperature, however, in the ferromagnetic state, the gap becomes almost zero. The other Gd pnictides are found to have a small overlap of the conduction band at the X point and the valence band at the Gamma point in the majority-spin channel. A small gap opens in the spin-minority channel of GdP and GdAs, which are thus half-metallic. This spin-minority gap closes in semimetallic GdSb and GdBi. While GdN is found to be ferromagnetic, the other Gd-pnictides are found to be antiferromagnetic, with ordering along [111]. From calculations with different magnetic configurations, a Heisenberg model with first and second nearest neighbor exchange parameters is extracted. The Heisenberg model is then used to predict Curie-Weiss and N'eel temperatures and critical magnetic fields within mean field and compared with experimental data. The trends are found to be in good agreement with the experimental data.
We give a proof that OCA holds in the Pmax extension of L(R). The proof is general enough to be adapted to most Pmax variations.