Fourth Order Gradient Symplectic Integrator Methods for Solving the Time-Dependent Schr\"odinger Equation

We show that the method of splitting the operator ${\rm e}^{\epsilon(T+V)}$ to fourth order with purely positive coefficients produces excellent algorithms for solving the time-dependent Schr\"odinger equation. These algorithms require knowing the potential and the gradient of the potential. One 4th order algorithm only requires four Fast Fourier Transformations per iteration. In a one dimensional scattering problem, the 4th order error coefficients of these new algorithms are roughly 500 times smaller than fourth order algorithms with negative coefficient, such as those based on the traditional Ruth-Forest symplectic integrator. These algorithms can produce converged results of conventional second or fourth order algorithms using time steps 5 to 10 times as large. Iterating these positive coefficient algorithms to 6th order also produced better converged algorithms than iterating the Ruth-Forest algorithm to 6th order or using Yoshida's 6th order algorithm A directly.Comment: 11 pages, 2 figures, submitted to J. Chem. Phy

Deliberative Agenda Setting: Piloting Reform of Direct Democracy in California

Can the people deliberate to set the agenda for direct democracy in large scale states? How might such an institution work? The 2011 California Deliberative Poll piloted a solution to this problem helping to produce proposals that went to the ballot and also to the legislature. The paper reports on how this pilot worked and what it suggests about a possible institution to solve the deliberative agenda setting problem. The legislative proposal passed the legislature but the ballot proposition (Prop 31) failed. However, we show that the proposals actually deliberated on by the people might well have passed if not encumbered by additional elements not deliberated on by the public that drew opposition. The paper ends with an outline of how the process of deliberative agenda setting for the initiative might work, vetting proposals once every two years that could get on the ballot for a greatly reduced cost in signature collections. Adding deliberation to the agenda setting process would allow for a thoughtful and informed public will formation to determine the agenda for direct democracy

Rank-finiteness for modular categories

We prove a rank-finiteness conjecture for modular categories: up to equivalence, there are only finitely many modular categories of any fixed rank. Our technical advance is a generalization of the Cauchy theorem in group theory to the context of spherical fusion categories. For a modular category $\mathcal{C}$ with $N=ord(T)$, the order of the modular $T$-matrix, the Cauchy theorem says that the set of primes dividing the global quantum dimension $D^2$ in the Dedekind domain $\mathbb{Z}[e^{\frac{2\pi i}{N}}]$ is identical to that of $N$.Comment: 25 pages (last version). Version 2: removed weakly integral rank 6 and integral rank 7 section, improved rank 5 classification up to monoidal equivalence. Version 3: removed rank 5 classification (note title change)--this will be published separately. Significantly improved expositio

Testing for positive edge responses in a fragmented landscape in the Eastern Tiger (Papilio glaucus) and the Spicebush (P. troilus) swallowtail butterflies

Landscape changes such as habitat fragmentation and habitat loss are contributing to a global decline in biodiversity. While habitat fragmentation research has mainly focused on species that avoid edges, or the boundaries between different landcover types (negative edge response), a hypothesized resource distribution model predicts that species that require complementary resources in different landcovers will be most abundant at edges (positive edge response). Adults of Eastern Tiger (Papilio glaucus) and Spicebush (P. troilus) swallowtail butterflies require forests for oviposition sites and meadows for nectar resources. I examined the relative abundance and flight orientation of both species in relation to the forest/meadow edge to evaluate their edge response. Overall, I found that their distribution and flight behaviour was consistent with the positive edge response model, however there were differences between species and sexes. My results suggest that some degree of forest fragmentation in southernwestern Ontario can actually benefit some native species

Extrapolated High-Order Propagators for Path Integral Monte Carlo Simulations

We present a new class of high-order imaginary time propagators for path-integral Monte Carlo simulations by subtracting lower order propagators. By requiring all terms of the extrapolated propagator be sampled uniformly, the subtraction only affects the potential part of the path integral. The negligible violation of positivity of the resulting path integral at small time steps has no discernable affect on the accuracy of our method. Thus in principle arbitrarily high order algorithms can be devised for path-integral Monte Carlo simulations. We verify this claim is by showing that fourth, sixth, and eighth order convergence can indeed be achieved in solving for the ground state of strongly interacting quantum many-body systems such as bulk liquid $^4$He.Comment: 9 pages and 3 figures. Submitted to J. Chem. Phy

On classification of modular categories by rank

The feasibility of a classification-by-rank program for modular categories follows from the Rank-Finiteness Theorem. We develop arithmetic, representation theoretic and algebraic methods for classifying modular categories by rank. As an application, we determine all possible fusion rules for all rank=$5$ modular categories and describe the corresponding monoidal equivalence classes.Comment: arXiv admin note: substantial text overlap with arXiv:1310.705

On Alexander Wylie’s Jottings on the Science of the Chinese Arithmetic

Oral Presentation(Theme 2): R102Starting from August of 1852 the British Protestant missionary and sinologist, Alexander Wylie (1815–1887), published in nine instalments an account Jottings on the Science of the Chinese Arithmetic in the newspaper North China Herald. He explained clearly the purpose of his account at the beginning: ‘The object of the following desultory notes, made from time to time, in the course of some researches entered upon, with another purpose in view, is to draw attention to the state of the arithmetical science in China, a subject which has not been so fully explored as it might with advantage, and on which some erroneous statements have been current in modern publications.’ Alexander Wylie is a well-known figure in the last quarter of the Qing Dynasty for his contribution in transmitting Western science into China during the latter half of the 19th century. In mathematics he was known for translating three treatises in collaboration with the Qing mathematician Li Shanlan (1811–1882) — Supplementary Elements of Geometry in 1856 but published in 1865 (believed to be based on the English translation of Book VII to XV of Elements by Henry Billingsley in 1570), Treatise of Algebra in 1859 (based on Elements of Algebra by Agustus De Morgan in 1835) and Analytical Geometry and Differential and Integral Calculus Step by Step in 1859 (based on Elements of Analytical Geometry and of the Differential and Integral Calculus of Elias Loomis in 1850). He was also the author of Compendium of Arithmetic published in 1853. This presentation will discuss the knowledge of Chinese science and mathematics which most European sinologists of the 18th and 19th centuries possessed and the low regard they held it in, but the viewpoint of which was critically examined by Wylie in his account.published_or_final_versio

Effective algebraic degeneracy

We prove that any nonconstant entire holomorphic curve from the complex line C into a projective algebraic hypersurface X = X^n in P^{n+1}(C) of arbitrary dimension n (at least 2) must be algebraically degenerate provided X is generic if its degree d = deg(X) satisfies the effective lower bound: d larger than or equal to n^{{(n+1)}^{n+5}}

Cyclin F Is Degraded during G2-M by Mechanisms Fundamentally Different from Other Cyclins

Cyclin F, a cyclin that can form SCF complexes and bind to cyclin B, oscillates in the cell cycle with a pattern similar to cyclin A and cyclin B. Ectopic expression of cyclin F arrests the cell cycle in G2/M. How the level of cyclin F is regulated during the cell cycle is completely obscure. Here we show that, similar to cyclin A, cyclin F is degraded when the spindle assembly checkpoint is activated and accumulates when the DNA damage checkpoint is activated. Cyclin F is a very unstable protein throughout much of the cell cycle. Unlike other cyclins, degradation of cyclin F is independent of ubiquitination and proteasome-mediated pathways. Interestingly, proteolysis of cyclin F is likely to involve metalloproteases. Rapid destruction of cyclin F does not require the N-terminal F-box motif but requires the COOH-terminal PEST sequences. The PEST region alone is sufficient to interfere with the degradation of cyclin F and confer instability when fused to cyclin A. These data show that although cyclin F is degraded at similar time as the mitotic cyclins, the underlying mechanisms are entirely distinct