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We give an adaptive algorithm which tests whether an unknown Boolean function f : {0, 1} n → {0, 1} is unate, i.e. every variable of f is either non-decreasing or non-increasing, or ε-far from unate with one-sided error using O(n 3/4 /ε 2 ) queries. This improves on the best adaptive O(n/ε)-query algorithm from Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova and Seshadhri [BCP + 17b] when 1/ε n 1/4 . Combined with the Ω(n)-query lower bound for non-adaptive algorithms with one-sided error of [CWX17, BCP + 17a], we conclude that adaptivity helps for the testing of unateness with one-sided error. A crucial component of our algorithm is a new subroutine for finding bi-chromatic edges in the Boolean hypercube called adaptive edge search.

We prove a lower bound ofΩ(n 1/3 ) for the query complexity of any two-sided and adaptive algorithm that tests whether an unknown Boolean function f : {0, 1} n → {0, 1} is monotone or far from monotone. This improves the recent bound ofΩ(n 1/4 ) for the same problem by Belovs and Blais [BB16]. Our result builds on a new family of random Boolean functions that can be viewed as a two-level extension of Talagrand's random DNFs.Beyond monotonicity, we also prove a lower bound ofΩ(n 2/3 ) for any two-sided and adaptive algorithm, and a lower bound ofΩ(n) for any one-sided and non-adaptive algorithm for testing unateness, a natural generalization of monotonicity. The latter matches the recent linear upper bounds by Khot and Shinkar [KS16] and by Chakrabarty and Seshadhri [CS16].

An essential component of any artificial pancreas is on the prediction of blood glucose levels as a function of exogenous and endogenous perturbations such as insulin dose, meal intake, and physical activity and emotional tone under natural living conditions. In this article, we present a new data-driven state-space dynamic model with time-varying coefficients that are used to explicitly quantify the time-varying patient-specific effects of insulin dose and meal intake on blood glucose fluctuations. Using the 3-variate time series of glucose level, insulin dose, and meal intake of an individual type 1 diabetic subject, we apply an extended Kalman filter (EKF) to estimate time-varying coefficients of the patient-specific state-space model. We evaluate our empirical modeling using (1) the FDA-approved UVa/Padova simulator with 30 virtual patients and (2) clinical data of 5 type 1 diabetic patients under natural living conditions. Compared to a forgetting-factor-based recursive ARX model of the same order, the EKF model predictions have higher fit, and significantly better temporal gain and J index and thus are superior in early detection of upward and downward trends in glucose. The EKF based state-space model developed in this article is particularly suitable for model-based state-feedback control designs since the Kalman filter estimates the state variable of the glucose dynamics based on the measured glucose time series. In addition, since the model parameters are estimated in real time, this model is also suitable for adaptive control.

The VSD can be used as a complementary tool to detect meal and estimate meal size in absence of a meal announcement.