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Background Type 2 diabetes (T2D) is characterized by co-morbid cardiometabolic abnormalities and elevated heart failure (HF) risk. Purpose The purpose of this study was to evaluate the associations of baseline and longitudinal changes in cardiometabolic health with risk of HF among adults with T2D. Methods Adults with T2D enrolled in the Look AHEAD (Action for Health in Diabetes) trial without prevalent HF were included. Adjusted Cox models were used to create a cardiometabolic health score incorporating target levels of parameters weighted based on relative risk for HF. The associations of baseline, 1- and 4-year changes in the cardiometabolic health score with risk of overall HF, HF with preserved ejection fraction (HFpEF; EF≥50%), and HF with reduced EF (HFrEF; EF Results Of the 5,080 participants included, there were 257 incident HF events during 12.4-year follow-up. The cardiometabolic health score included 2-points each for target levels of waist circumference, glomerular filtration rate, urine albumin-to-creatinine ratio and 1-point each for blood pressure and hemoglobin A1c at target (Table 1). Higher baseline cardiometabolic health score was significantly associated with lower risk of overall HF (adjusted hazard ratio [aHR] per 1-unit higher score, 0.72 [95% CI, 0.66–0.79]) with similar associations observed for HFpEF and HFrEF (Table 2). Improvement in cardiometabolic health over 1- and 4-years was significantly associated with lower risk of overall HF (aHR per 1-unit increase in score at 1- and 4-years, 0.82 [95% CI, 0.73–0.92] and 0.80 [95% CI, 0.70–0.91], respectively). Conclusions Among adults with T2D, achieving target levels of more cardiometabolic health parameters at baseline and sustained improvements were associated with lower HF risk. Funding Acknowledgement Type of funding sources: Public grant(s) – National budget only. Main funding source(s): National Institute of Diabetes and Digestive and Kidney Diseases

A model of a stochastic froth is introduced in which the rate of random coalescence of a pair of bubbles depends on an inverse power law of their sizes. The main question of interest is whether froths with a large number of bubbles can grow in a stable fashion; that is, whether under some time-varying change of scale the distributions of rescaled bubble sizes become approximately stationary. It is shown by way of a law of large numbers for the froths that the question can be re-interpreted in terms of a measure flow solving a nonlinear Boltzmann equation that represents an idealized deterministic froth. Froths turn out to be stable in the sense that there are scalings in which the rescaled measure flow is tight and, for a particular case, stable in the stronger sense that the rescaled flow converges to an equilibrium measure. Precise estimates are also given for the degree of tightness of the rescaled measure flows.

For steering a linear stochastic system with bounded controls so as to be as close as possible to a given hyperplane at the terminal time, Benes proved that the controller that predicts which side of the hyperplane the state would be on with no further control, and then applies full “bang” in the appropriate direction, optimal, a result conjectured y Hilborn. In this paper a different proof of this result is presented, using martingale-based optimality criteria.

A recursive algorithm for estimating the constant but unknown parameters of a controlled ARMA process is presented. The algorithm is a recursive version of an off-line algorithm using three stages of standard least-squares. In the first stage the parameters of a controlled AR model of degree p are estimated. The residuals used in this stage are employed in the second stage to estimate the parameters of a controlled ARMA process. The first two stages constitute a recursive version of Durbin's algorithm. The model obtained in the second stage is used to filter the input, output and residuals and these filtered variables are used in the final stage to obtain improved estimates of the controlled ARMA process. It is shown that the estimate is (globally) p-consistent, i.e. that the estimate converges a.s. as the number of data tends to infinity, to a vector which, in turn, converges to the true parameter vector as the degree p of the AR model tends to infinity.

All sets mentioned in this paper will be assumed to be appropriately measurable without further comment, and events of probability zero will be neglected. We start with a fixed (large, convex) subset R of the plane, termed the “retina”.

These notes introduce the mathematical apparatus that is relevant for an understanding and precise description of the idea of a dynamical system driven by white noise. The main tools are the stochastic integral and stochastic differential equations of Ito; however the representations of Fisk and Stratonovich are also included, not only because they have a nice physical interpretation, but because they seem to provide a natural formalism for processes that lie on manifolds. The treatment is sketchy in places but we have tried to give at least the gist of the main arguments. Basic references are Ito [1,2,3] , McKean[4] and Wong [5].

It is known that any functional of Brownian motion with finite second moment can be expressed as the sum of a constant and an Ito stochastic integral. It is also known that homogeneous additive functionals of Brownian motion with finite expectations have a similar representation. This paper extends these results in several ways. It is shown that any finite functional of Brownian motion can be represented as a stochastic integral. This representation is not unique, but if the functional has a finite expectation it does have a unique representation as a constant plus a stochastic integral in which the process of indefinite integrals is a martingale. A corollary of this result is that any martingale (on a closed interval) that is measurable with respect to the increasing family of $\sigma$-fields generated by a Brownian motion is equal to a constant plus an indefinite stochastic integral. Sufficiently well-behaved Frechet-differentiable functionals have an explicit representation as a stochastic integral in which the integrand has the form of conditional expectations of the differential.

Dr. J. Alan Steppe has kindly pointed out two serious flaws, and some others less serious, in the proof of our paper [2]. However these can all be corrected, so that the result remains valid.