A Is Not For Ally

Most people can recall their first crush. They think fondly back to age ten or eleven when they first “went boy-crazy” or couldn’t focus on sixth-grade English because that cute girl was in their class. This did not happen for me. I do, however, vividly remember it happening for everyone around me. [excerpt

Fractional Chemotaxis Diffusion Equations

We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modelling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or macro-molecular crowding. The mesoscopic models are formulated using Continuous Time Random Walk master equations and the macroscopic models are formulated with fractional order differential equations. Different models are proposed depending on the timing of the chemotactic forcing. Generalizations of the models to include linear reaction dynamics are also derived. Finally a Monte Carlo method for simulating anomalous subdiffusion with chemotaxis is introduced and simulation results are compared with numerical solutions of the model equations. The model equations developed here could be used to replace Keller-Segel type equations in biological systems with transport hindered by traps, macro-molecular crowding or other obstacles.Comment: 25page

Lower central series and free resolutions of hyperplane arrangements

If $M$ is the complement of a hyperplane arrangement, and A=H^*(M,\k) is the cohomology ring of $M$ over a field of characteristic 0, then the ranks, $\phi_k$, of the lower central series quotients of $\pi_1(M)$ can be computed from the Betti numbers, b_{ii}=\dim_{\k} \Tor^A_i(\k,\k)_i, of the linear strand in a (minimal) free resolution of \k over $A$. We use the Cartan-Eilenberg change of rings spectral sequence to relate these numbers to the graded Betti numbers, b'_{ij}=\dim_{\k} \Tor^E_i(A,\k)_j, of a (minimal) resolution of $A$ over the exterior algebra $E$. From this analysis, we recover a formula of Falk for $\phi_3$, and obtain a new formula for $\phi_4$. The exact sequence of low degree terms in the spectral sequence allows us to answer a question of Falk on graphic arrangements, and also shows that for these arrangements, the algebra $A$ is Koszul iff the arrangement is supersolvable. We also give combinatorial lower bounds on the Betti numbers, $b'_{i,i+1}$, of the linear strand of the free resolution of $A$ over $E$; if the lower bound is attained for $i = 2$, then it is attained for all $i \ge 2$. For such arrangements, we compute the entire linear strand of the resolution, and we prove that all components of the first resonance variety of $A$ are local. For graphic arrangements (which do not attain the lower bound, unless they have no braid sub-arrangements), we show that $b'_{i,i+1}$ is determined by the number of triangles and $K_4$ subgraphs in the graph.Comment: 25 pages, to appear in Trans. Amer. Math. So

Resonance, linear syzygies, Chen groups, and the Bernstein-Gelfand-Gelfand correspondence

If \A is a complex hyperplane arrangement, with complement X, we show that the Chen ranks of G=\pi_1(X) are equal to the graded Betti numbers of the linear strand in a minimal, free resolution of the cohomology ring A=H^*(X,\k), viewed as a module over the exterior algebra E on \A: \theta_k(G) = \dim_\k Tor^E_{k-1}(A,\k)_k, where \k is a field of characteristic 0, and k\ge 2. The Chen ranks conjecture asserts that, for k sufficiently large, \theta_k(G) =(k-1) \sum_{r\ge 1} h_r \binom{r+k-1}{k}, where h_r is the number of r-dimensional components of the projective resonance variety R^1(\A). Our earlier work on the resolution of A over E and the above equality yield a proof of the conjecture for graphic arrangements. Using results on the geometry of R^1(\A) and a localization argument, we establish the conjectured lower bound for the Chen ranks of an arbitrary arrangement \A. Finally, we show that there is a polynomial P(t) of degree equal to the dimension of R^1(\A), such that \theta_k(G) = P(k), for k sufficiently large.Comment: 21 pages; final versio

Fractional Fokker-Planck Equations for Subdiffusion with Space-and-Time-Dependent Forces

We have derived a fractional Fokker-Planck equation for subdiffusion in a general space-and- time-dependent force field from power law waiting time continuous time random walks biased by Boltzmann weights. The governing equation is derived from a generalized master equation and is shown to be equivalent to a subordinated stochastic Langevin equation.Comment: 5 page

Fractional chemotaxis diffusion equations

We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modeling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or macromolecular crowding. The mesoscopic models are formulated using continuous time random walk equations and the macroscopic models are formulated with fractional order differential equations. Different models are proposed depending on the timing of the chemotactic forcing. Generalizations of the models to include linear reaction dynamics are also derived. Finally a Monte Carlo method for simulating anomalous subdiffusion with chemotaxis is introduced and simulation results are compared with numerical solutions of the model equations. The model equations developed here could be used to replace Keller-Segel type equations in biological systems with transport hindered by traps, macromolecular crowding or other obstacles

Distribution of Mutual Information

The mutual information of two random variables i and j with joint probabilities t_ij is commonly used in learning Bayesian nets as well as in many other fields. The chances t_ij are usually estimated by the empirical sampling frequency n_ij/n leading to a point estimate I(n_ij/n) for the mutual information. To answer questions like "is I(n_ij/n) consistent with zero?" or "what is the probability that the true mutual information is much larger than the point estimate?" one has to go beyond the point estimate. In the Bayesian framework one can answer these questions by utilizing a (second order) prior distribution p(t) comprising prior information about t. From the prior p(t) one can compute the posterior p(t|n), from which the distribution p(I|n) of the mutual information can be calculated. We derive reliable and quickly computable approximations for p(I|n). We concentrate on the mean, variance, skewness, and kurtosis, and non-informative priors. For the mean we also give an exact expression. Numerical issues and the range of validity are discussed.Comment: 8 page