. We introduce the ergodic complex plane, a global analogue of Hilger's local complex plane on time scales, which simultaneously encodes exponential growth and frequency. Averaging the cylinder transformation on a periodic time scale leads to the notions of ergodic growth rate and ergodic frequency, unifying local and global stability perspectives. This yields the ergodic cylinder transformation, a univalent map inducing an orthogonal curvilinear coordinate system on the regressive complex plane. Within this framework, we develop a decomposition analogous to Hilger's real and imaginary parts, and define the box plus operation, extending the circle plus operation globally.

Similarity transformations of linear systems of differential equations are briefly discussed. The companion form of a matrix and of the system is introduced as well as the fundamental control concept of observability. A condition for linear systems that guarantees the existence and uniqueness of a linear transformation that transforms the original system into a companion system is stated. The question is then raised: if there is no way to transform a given system into its companion form, is there at least a way to transform the system matrix into a companion form? The concluding theorem proves that the answer is yes, and furthermore any transformation matrix that will do so must be a full rank observability matrix of the system. This result holds for all systems whose matrix’s characteristic polynomial is equal to its minimal polynomial.

The Gibbs phenomena associated to partial sums of Fourier series are now well understood. In this paper, we show that Gibbs phenomena also occur for expansions of functions in terms of members of one of several general classes of orthogonal polynomials, in particular treating in a relatively unified manner expansions in either Legendre (more generally Jacobi), Hermite, or Laguerre polynomials. Noteworthy is the fact that the Gibbs constants associated to all of these expansions have the same value, approximately 0.0893 or more precisely 1π∫0πsin⁡ttdt−12.

We examine the region of convergence (ROC) of the Laplace transform on time scales. An ergodic approach is employed to both calculate the ROC and the inversion of the transform. It turns out that the problem of computing the ROC is intimately tied with the problem of computing regions of stability for the time scale exponential function ez(t,t0). The latter is a problem for which the current authors have used ergodic techniques to arrive at a solution, and so in the current paper we will expound upon these notions and show how they can be used to think about the Laplace transform in an effective and efficient manner.

Disturbance shapes the structure and function of aquatic communities and ecosystems, but the dynamics of ice are a less studied dimension of the disturbance-regime of rivers. We investigated effects of a river-ice regime on organic-matter dynamics and feeding ecology of aquatic insects. Samples of biofilm and aquatic insects for gut content analysis were collected monthly from Big Creek, a sixth - order tributary of the Middle Fork Salmon River in central Idaho, USA, during winter 2010-2011. Our results indicate that river ice affects both quantity and quality of organic matter available to, and used by, consumers. Specifically, scour from December and February ice break-up events reduced biofilm biomass by one-half and one-third, respectively, whereas quality (chlorophyll-a: ash-free dry mass) increased. Diets of scrapers, Rhithrogena (Heptageniidae) and Bibiocephala (Blephariceridae), collector-gatherers, Baetis (Baetidae), and collector-filterers, Simulium (Simulidae) appeared to follow patterns of organic matter. Following ice break-up events, diets of these taxa had increased proportions of diatom frustules, which are high-quality food resources due to their relatively high nutrient content. Other taxa, such as collector-gatherers, non-Tanypodinae (Chironomidae), and the collector-filterer, Arctopsyche grandis (Hydropsychidae), consistently consumed high proportions of diatom frustules and insect material, respectively, suggesting they were able to feed more selectively throughout winter. Our study indicates that ice regimes in temperate rivers can affect organic-matter dynamics and feeding ecology of aquatic insects, a possibility that deserves additional investigation, particularly in light of potential changes to the ice regimes of rivers with changing climate.

Background Rapid advances in the past decade have shown that dysbiosis of the gut microbiome is a key hallmark of rheumatoid arthritis (RA). Yet, the relationship between gut microbiome and clinical improvement in RA disease activity remains unclear. In this study, we explored the gut microbiome of patients with RA to identify features that are associated with, as well as predictive of, minimum clinically important improvement (MCII) in disease activity. Methods Whole metagenome shotgun sequencing was performed on 64 stool samples, which were collected from 32 patients with RA at two separate time-points. The Clinical Disease Activity Index (CDAI) of each patient was measured at both time-points to assess achievement of MCII; depending on this clinical status, patients were distinguished into two groups. Multiple linear regression models were used to identify microbial taxa and biochemical pathways associated with MCII while controlling for potentially confounding factors. Lastly, a deep-learning neural network was trained upon gut microbiome, clinical, and demographic data at baseline to classify patients according to MCII status, thereby enabling the prediction of whether a patient will achieve MCII at follow-up. Results We determined that MCII status can explain a significant proportion of the overall compositional variance in the gut microbiome (R2 = 3.8%, P = 0.005, PERMANOVA). Additionally, by looking at patients baseline gut microbiome profiles, we observed significantly different microbiome traits between patients who eventually showed MCII and those who did not. Taxonomic features include alpha- and beta-diversity measures, as well as several microbial taxa, such as Coprococcus, Bilophila sp. 4_1_30, and Ruminococcus sp. Functional profiling identified thirteen biochemical pathways, most of which were involved in the biosynthesis of L-arginine and L-methionine, to be differentially abundant between the MCII patient groups. In addition to these observations at baseline, we found microbiome features that vary differently in fold-change (from baseline to follow-up) between the two patient groups. These results could suggest that, depending on the clinical course, gut microbiomes not only start at different ecological states, but also are on separate trajectories. Finally, the neural network proved to be highly effective in predicting which patient will achieve MCII (balanced accuracy = 90.0%), demonstrating potential clinical utility of gut microbiome profiles. Conclusions Our findings confirm the presence of taxonomic and functional signatures of the gut microbiome associated with MCII in RA patients. Ultimately, the gut microbiome may aid in the development of non-invasive tools for predicting future prognosis in RA.

We present a new technique for calculating the Pötzsche, Siegmund, and Wirth's region of exponential stability for a time scale, Sℂ(T). The approach taken here unifies and extends other results on this topic in the literature. We establish a tractable algorithm for computing Sℂ(T) in terms of the asymptotic extended graininesses of the time scale and their relative frequencies.

Alfaro-Maguyon, Maria1; Anderson, Tiffany1; Davis, John1; Hothem, Zachary1; Ghita, Gabriela1; Wang, Zhongkai1; Brumback, Babette1; Walters, Michael1; Vanzant, Erin1; Meerkov, Meir1; Rosenthal, Martin1; Stephen Smith, R.1; Croft, Chasen1; Moore, Frederick1; Marker, Peggy1; Brakenridge, Scott1; Mohr, Alicia1; Efron, Philip1 Author Information

We consider dynamic systems which evolve on discrete time domains where the time steps form a sequence of independent, identically distributed random variables. In particular, we classify the mean-square stability of linear systems on these time domains using quadratic Lyapunov functionals. In the case where the system matrix is a function of the time step, our results agree with and generalize stability results found in the Markov jump linear systems literature. In the case where the system matrix is constant, our results generalize, illuminate, and extend to the stochastic realm results in the field of dynamic equations on time scales. In order to help see the factors that contribute to stability, we prove a sufficient condition for the solvability of the Lyapunov equation by appealing to a fixed point theorem of Ran and Reurings. Finally, an example using observer-based feedback control is presented to demonstrate the utility of the results to control engineers who cannot guarantee uniform timing of the system.

Ecosystems are connected through fluxes of nutrients, organic matter, and organisms. Disturbances that alter structure and function of one ecosystem may have consequences for other linked ecosystems. We investigated how wildfire and subsequent debris flows altered fluxes of invertebrates from tributaries in the Salmon River Basin, Idaho, USA. We compared fluxes of invertebrates downstream through drift and laterally via insect emergence from streams with varying disturbance histories (unburned, burned, and burned + debris flow) during two summers (3–4 years post fire and 2–3 years post debris flow). We observed that the combination of wildfire and debris flow increased the biomass export of invertebrates from tributaries to main-stem ecosystems 2–3 × compared to other streams. In contrast, aquatic insect emergence did not differ in magnitude among streams of different disturbance histories, but instead diverged in timing. Underwater surveys indicated trout in the main-stem river selected confluence habitats, with a tendency for stronger selection of confluences with burned streams. In a behavioral comparison between confluence and non-confluence habitats, rates of agonistic behavior were 4–6 × higher in confluence areas, indicating that confluences may be worth defending. Abundances of web-spinning spiders that depend on emerging insects did not vary with disturbance history in early-mid summer, but tended to be highest in riparian areas along burned streams by August. Because wildfire and debris flows are predicted to increase, our results elucidate potential pathways by which altered disturbance regimes may affect coupled aquatic-terrestrial ecosystems.

Little information is available regarding standards and processes for identifying and evaluating Stage D heart failure (HF) patients for advanced therapies, posing challenges for clinical care, quality improvement and research. In December 2012, we implemented a formalized informed consent and advanced therapies evaluation process facilitated by a dedicated nurse coordinator. Here we describe the population consented for evaluation and the outcomes of this process.

This paper addresses the optimal regulator problem for dynamic systems which evolve on discrete time domains where the time steps form a sequence of independent, identically distributed random variables. We introduce generalizations of the dynamic and algebraic Riccati equations and use them to find optimal gain matrices in both the finite-horizon and infinite-horizon case. Examples are presented.

In this paper, the stability of a certain class of time-varying systems evolving over nonuniformly spaced discrete domains is analyzed. Switched systems, used here in the context of dynamic equations over time scale domains, arise naturally in applications when a continuous time system is discretized via a sample-and-hold method with multiple sample rates. The stability of switched systems is typically deduced by appealing to certain interrelated properties of the subsystems (such as pairwise commutativity [24], simultaneous diagonalization [7], simultaneous triangularizability [8], or other Lie algebraic conditions [1]) which imply the existence of a common quadratic Lyapunov function. A novel approach is used here to determine the existence of quadratic Lyapunov functions which does not rely on how the subsystems interact with each other. This new method instead examines the role they each play in the aggregate system by way of singular value conditions. Results implying switched system stability and instability are developed under the two primary methods of examining such systems: how the system behaves during arbitrary switching and how it behaves under the influence of a particular switching signal. Several examples illuminating the results are provided throughout the text, as well as important insight discussing certain bounds on these results as they apply to stability theory in general.