The Galois Complexity of Graph Drawing: Why Numerical Solutions are Ubiquitous for Force-Directed, Spectral, and Circle Packing Drawings

Many well-known graph drawing techniques, including force directed drawings, spectral graph layouts, multidimensional scaling, and circle packings, have algebraic formulations. However, practical methods for producing such drawings ubiquitously use iterative numerical approximations rather than constructing and then solving algebraic expressions representing their exact solutions. To explain this phenomenon, we use Galois theory to show that many variants of these problems have solutions that cannot be expressed by nested radicals or nested roots of low-degree polynomials. Hence, such solutions cannot be computed exactly even in extended computational models that include such operations.Comment: Graph Drawing 201

Experimental study of laboratory-heated CM2 chondrites Mighei and Murchison

We conducted experimental heating of two CM2 chondrites, Murchison and Mighei, to study changes in their oxygen isotopic compositions and mineralogy and explore possible genetic relationships between MCCs and normal CMs

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

Hard Instances of the Constrained Discrete Logarithm Problem

The discrete logarithm problem (DLP) generalizes to the constrained DLP, where the secret exponent $x$ belongs to a set known to the attacker. The complexity of generic algorithms for solving the constrained DLP depends on the choice of the set. Motivated by cryptographic applications, we study sets with succinct representation for which the constrained DLP is hard. We draw on earlier results due to Erd\"os et al. and Schnorr, develop geometric tools such as generalized Menelaus' theorem for proving lower bounds on the complexity of the constrained DLP, and construct sets with succinct representation with provable non-trivial lower bounds

Near-optimal mean value estimates for multidimensional Weyl sums

We obtain sharp estimates for multidimensional generalisations of Vinogradov's mean value theorem for arbitrary translation-dilation invariant systems, achieving constraints on the number of variables approaching those conjectured to be the best possible. Several applications of our bounds are discussed

Disposition of docosahexaenoic acid-paclitaxel, a novel taxane, in blood: in vitro and clinical pharmacokinetic studies

PURPOSE: Docosahexaenoic acid-paclitaxel is as an inert prodrug composed of the natural fatty acid DHA covalently linked to the C2'-position of paclitaxel (M. O. Bradley et al., Clin. Cancer Res., 7: 3229-3238, 2001). Here, we examined the role of protein binding as a determinant of the pharmacokinetic behavior of DHA-paclitaxel. EXPERIMENTAL DESIGN: The blood distribution of DHA-paclitaxel was studied in vitro using equilibrium dialysis and in 23 cancer patients receiving the drug as a 2-h i.v. infusion (dose, 200-1100 mg/m(2)). RESULTS: In vitro, DHA-paclitaxel was found to bind extensively to human plasma (99.6 +/- 0.057%). The binding was concentration independent (P = 0.63), indicating a nonspecific, nonsaturable process. The fraction of unbound paclitaxel increased from 0.052 +/- 0.0018 to 0.055 +/- 0.0036 (relative increase, 6.25%; P = 0.011) with an increase in DHA-paclitaxel concentration (0-1000 microg/ml), suggesting weakly competitive drug displacement from protein-binding sites. The mean (+/- SD) area under the curve of unbound paclitaxel increased nonlinearly with dose from 0.089 +/- 0.029 microg.h/ml (at 660 mg/m(2)) to 0.624 +/- 0.216 microg.h/ml (at 1100 mg/m(2)), and was associated with the dose-limiting neutropenia in a maximum-effect model (R(2) = 0.624). A comparative analysis indicates that exposure to Cremophor EL and unbound paclitaxel after DHA-paclitaxel (at 1100 mg/m(2)) is similar to that achieved with paclitaxel on clinically relevant dose schedules. CONCLUSIONS: Extensive binding to plasma proteins may explain, in part, the unique pharmacokinetic profile of DHA-paclitaxel described previously with a small volume of distribution ( approximately 4 liters) and slow systemic clearance ( approximately 0.11 liters/h)

Tricritical Behavior of Two-Dimensional Scalar Field Theories

We compute by Monte Carlo numerical simulations the critical exponents of two-dimensional scalar field theories at the $\lambda\phi^6$ tricritical point. The results are in agreement with the Zamolodchikov conjecture based on conformal invariance.Comment: 13 pages, uuencode tar-compressed Postscript file, preprint numbers: IF/UFRJ/25/94, DFTUZ 94.06 and NYU--TH--94/10/0