From fractions to complete Segal spaces

We show that the Rezk classification diagram of a relative category admitting a homotopical version of the two-sided calculus of fractions is a Segal space up to Reedy-fibrant replacement. This generalizes the result of Rezk and Bergner on the classification diagram of a closed model category, as well as the result of Barwick and Kan on the classification diagram of a partial model category.Comment: 21 pages, LaTeX. Changes in v3: added some expository material, following suggestions by anonymous referee. (N.B. numbering has changed.

Equilateral sets in the ℓ1 sum of Euclidean spaces

Let En denote the (real) n-dimensional Euclidean space. It is not known whether an equilateral set in the ℓ1 sum of Ea and Eb , denoted here as Ea⊕1Eb , has maximum size at least dim(Ea⊕1Eb)+1=a+b+1 for all pairs of a and b. We show, via some explicit constructions of equilateral sets, that this holds for all a⩽27 , as well as some other instances

Near-Optimal Time and Sample Complexities for Solving Discounted Markov Decision Process with a Generative Model

In this paper we consider the problem of computing an $\epsilon$-optimal policy of a discounted Markov Decision Process (DMDP) provided we can only access its transition function through a generative sampling model that given any state-action pair samples from the transition function in $O(1)$ time. Given such a DMDP with states $S$, actions $A$, discount factor $\gamma\in(0,1)$, and rewards in range $[0, 1]$ we provide an algorithm which computes an $\epsilon$-optimal policy with probability $1 - \delta$ where \emph{both} the time spent and number of sample taken are upper bounded by $$O\left[\frac{|S||A|}{(1-\gamma)^3 \epsilon^2} \log \left(\frac{|S||A|}{(1-\gamma)\delta \epsilon} \right) \log\left(\frac{1}{(1-\gamma)\epsilon}\right)\right] ~.$$ For fixed values of $\epsilon$, this improves upon the previous best known bounds by a factor of $(1 - \gamma)^{-1}$ and matches the sample complexity lower bounds proved in Azar et al. (2013) up to logarithmic factors. We also extend our method to computing $\epsilon$-optimal policies for finite-horizon MDP with a generative model and provide a nearly matching sample complexity lower bound.Comment: 31 pages. Accepted to NeurIPS, 201

Phase resetting reveals network dynamics underlying a bacterial cell cycle

Genomic and proteomic methods yield networks of biological regulatory interactions but do not provide direct insight into how those interactions are organized into functional modules, or how information flows from one module to another. In this work we introduce an approach that provides this complementary information and apply it to the bacterium Caulobacter crescentus, a paradigm for cell-cycle control. Operationally, we use an inducible promoter to express the essential transcriptional regulatory gene ctrA in a periodic, pulsed fashion. This chemical perturbation causes the population of cells to divide synchronously, and we use the resulting advance or delay of the division times of single cells to construct a phase resetting curve. We find that delay is strongly favored over advance. This finding is surprising since it does not follow from the temporal expression profile of CtrA and, in turn, simulations of existing network models. We propose a phenomenological model that suggests that the cell-cycle network comprises two distinct functional modules that oscillate autonomously and couple in a highly asymmetric fashion. These features collectively provide a new mechanism for tight temporal control of the cell cycle in C. crescentus. We discuss how the procedure can serve as the basis for a general approach for probing network dynamics, which we term chemical perturbation spectroscopy (CPS)

Extension of Public Warning System

This disclosure describes techniques to relay public warning system (PWS) messages from a host device to user devices that cannot receive broadcast messages from a cellular network. A PWS message broadcast by a cellular network is received by the host device and relayed to user devices within a personal area network (PAN). Fields from the incoming PWS message are compared to previously received PWS messages to avoid duplication of messages. Relay of the PWS message by the router enables reception of the PWS message by user devices in the PAN that are not connected to the cellular network. User configurable settings enable users to select or disable the feature of reception of PWS via the PAN