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We examine the acoustic characteristics of clear and plain conversational productions of Mandarin tones. Twenty-one native Mandarin speakers were asked to produce a selection of Mandarin words in both plain and clear speaking styles. Several tokens were gathered for each of the four tones giving a total of 2045 productions. Six critical tonal cues were computed for each production: fundamental frequency (F0) mean, slope, and second derivative, duration, mean intensity, and a binary variable coding whether the production involved creaky voice. A linear mixed-effects regression model was used to explore how these cues changed with respect to the clear versus plain distinction for each tone, with speaking style as the fixed effect and speaker being a random effect. The strongest effects detected were that duration and mean intensity increased in clear speech across speakers and tones. Tones 2 and 3 increased in mean F0 and Tone 4 increased its slope. An additional finding was that, for contour tones, speakers accomplished the increase in duration by stretching out the tone contours in time while largely not changing the F0 range. These results are discussed in terms of signal-based (affecting all tones) and code-based (enhancing contrast between tones) change.

Clear speech is a vocal style used when a speaker wishes to improve comprehension, usually due to the presence of external noise, less-than-optimal listening conditions, or when they are simply instructed to speak clearly. Clear speech has many distinguishing features, including increased duration, pitch, and amplitude, as well as the exaggeration of articulatory movement. We use game theory to model the phenomenon of clear speech, and make predictions of how it changes under different circumstances. We view the behaviours of speakers and hearers when communicating as optimal strategies in communication games. When comprehension becomes more difficult, the optimal strategies of the games shift so that speakers exert more energy to improve the likelihood of accurate communication. We discuss how our models correspond to experimental observations and see what predictions are made for future experiments.

The general theory developed by Ben Yaacov for metric structures provides Fraïssé limits which are approximately ultrahomogeneous. We show here that this result can be strengthened in the case of relational metric structures. We give an extra condition that guarantees exact ultrahomogenous limits. The condition is quite general. We apply it to stochastic processes, the class of diversities, and its subclass of $L_1$ diversities.

Contact tracing has played a central role in COVID-19 control in many jurisdictions and is often used in conjunction with other measures such as travel restrictions and social distancing mandates. Contact tracing is made ineffective, however, by delays in testing, calling, and isolating. Even if delays are minimized, contact tracing triggered by testing of symptomatic individuals can only prevent a fraction of onward transmissions from contacts. Without other measures in place, contact tracing alone is insufficient to prevent exponential growth in the number of cases in a population with little immunity. Even when used effectively with other measures, occasional bursts in call loads can overwhelm contact tracing systems and lead to a loss of control. We propose embracing approaches to COVID-19 contact tracing that broadly test individuals without symptoms, in whatever way is economically feasible—either with fast and cheap tests that can be deployed widely, with pooled testing, or with screening of judiciously chosen groups of high-risk individuals. These considerations are important both in regions where widespread vaccination has been deployed and in those where few residents have been immunized.

COVID-19 remains a major public health concern, with large resurgences even where there has been widespread uptake of vaccines. Waning immunity and the emergence of new variants will shape the long-term burden and dynamics of COVID-19. We explore the transition to the endemic state, and the endemic incidence in British Columbia (BC), Canada and South Africa (SA), to compare low and high vaccination coverage settings with differing public health policies, using a combination of modelling approaches. We compare reopening (relaxation of public health measures) gradually and rapidly as well as at different vaccination levels. We examine how the eventual endemic state depends on the duration of immunity, the rate of importations, the efficacy of vaccines and the transmissibility. These depend on the evolution of the virus, which continues to undergo selection. Slower reopening leads to a lower peak level of incidence and fewer overall infections in the wave following reopening: as much as a 60% lower peak and a 10% lower total in some illustrative simulations; under realistic parameters, reopening when 70% of the population is vaccinated leads to a large resurgence in cases. The long-term endemic behaviour may stabilize as late as January 2023, with further waves of high incidence occurring depending on the transmissibility of the prevalent variant, duration of immunity, and antigenic drift. We find that long term endemic levels are not necessarily lower than current pandemic levels: in a population of 100,000 with representative parameter settings (Reproduction number 5, 1-year duration of immunity, vaccine efficacy at 80% and importations at 3 cases per 100K per day) there are over 100 daily incident cases in the model. Predicted prevalence at endemicity has increased more than twofold after the emergence and spread of Omicron. The consequent burden on health care systems depends on the severity of infection in immunized or previously infected individuals.

Estimating key aspects of transmission is crucial in infectious disease control. Serial intervals – the time between symptom onset in an infector and infectee – are fundamental, and help to define rates of transmission, estimates of reproductive numbers, and vaccination levels needed to prevent transmission. However, estimating the serial interval requires knowledge of individuals’ contacts and exposures (who infected whom), which is typically obtained through resource-intensive contact tracing efforts. We develop an alternate framework that uses virus sequences to inform who infected whom and thereby estimate serial intervals. The advantages are many-fold: virus sequences are often routinely collected to support epidemiological investigations and to monitor viral evolution. The genomic approach offers high resolution and cluster-specific estimates of the serial interval that are comparable with those obtained from contact tracing data. Our approach does not require contact tracing data, and can be used in large populations and over a range of time periods. We apply our techniques to SARS-CoV-2 sequence data from the first two waves of COVID-19 in Victoria, Australia. We find that serial interval estimates vary between clusters, supporting the need to monitor this key parameter and use updated estimates in onward applications. Compared to an early published serial interval estimate, using cluster-specific serial intervals can cause estimates of the effective reproduction number Rt to vary by a factor of up to 2–3. We also find that serial intervals estimated in settings such as schools and meat processing/packing plants tend to be shorter than those estimated in healthcare facilities.

Significance We provide a simple model of COVID-19 transmission at workplaces, events, and other settings. We use data from reported single-event, short-duration outbreaks to estimate the transmission rate, number of contacts, and turnover at events. We use these to predict how many new infections are expected to occur at various events given the presence of a single infectious individual. We then determine which types of interventions will be the most effective in reducing the number of infections: reducing transmission rates (such as with masks), social distancing (reducing the number of people in contact), or bubbling (keeping contact groups small and consistent)., COVID-19 is a global pandemic with over 25 million cases worldwide. Currently, treatments are limited, and there is no approved vaccine. Interventions such as handwashing, masks, social distancing, and “social bubbles” are used to limit community transmission, but it is challenging to choose the best interventions for a given activity. Here, we provide a quantitative framework to determine which interventions are likely to have the most impact in which settings. We introduce the concept of “event R,” the expected number of new infections due to the presence of a single infectious individual at an event. We obtain a fundamental relationship between event R and four parameters: transmission intensity, duration of exposure, the proximity of individuals, and the degree of mixing. We use reports of small outbreaks to establish event R and transmission intensity in a range of settings. We identify principles that guide whether physical distancing, masks and other barriers to transmission, or social bubbles will be most effective. We outline how this information can be obtained and used to reopen economies with principled measures to reduce COVID-19 transmission.

Diversities are a generalization of metric spaces in which a non-negative value is assigned to all finite subsets of a set, rather than just to pairs of points. Here we provide an analogue of the theory of negative-type metrics for diversities. We introduce negative-type diversities, and show that, as in the metric space case, they are a generalization of $$L_1$$ -embeddable diversities. We provide a number of characterizations of negative-type diversities, including a geometric characterization. Much of the recent interest in negative-type metrics stems from the connections between metric embeddings and approximation algorithms. We extend some of this work into the diversity setting, showing that lower bounds for embeddings of negative-type metrics into $$L_1$$ can be extended to diversities by using recently established extremal results on hypergraphs.

COVID-19 remains a major public health concern, with large resurgences even where there has been widespread uptake of vaccines. Waning immunity and the emergence of new variants will shape the long-term burden and dynamics of COVID-19. We explore the transition to the endemic state, and the endemic incidence, using a combination of modelling approaches. We compare gradual and rapid reopening and reopening at different vaccination levels. We examine how the eventual endemic state depends on the duration of immunity, the rate of importations, the efficacy of vaccines and the transmissibility. These depend on the evolution of the virus, which continues to undergo selection. Slower reopening leads to a lower peak level of incidence and fewer overall infections: as much as a 60% lower peak and a 10% lower total in some illustrative simulations; under realistic parameters, reopening when 70% of the population is vaccinated leads to a large resurgence in cases. The long-term endemic behaviour may stabilize as late as January 2023, with further waves of high incidence occurring depending on the transmissibility of the prevalent variant, duration of immunity, and antigenic drift. We find that long term endemic levels are not necessarily lower than current pandemic levels: in a population of 100,000 with representative parameter settings (Reproduction number 5, 1-year duration of immunity, vaccine efficacy at 80% and importations at 3 cases per 100K per day) there are over 100 daily incident cases in the model. The consequent burden on health care systems depends on the severity of infection in immunized or previously infected individuals.

The role of schools in the spread of the COVID-19 pandemic is controversial, with some claiming they are an important driver of the pandemic and others arguing that transmission in schools is negligible. School cluster reports that have been collected in various jurisdictions are a source of data about transmission in schools. These reports consist of the name of a school, a date, and the number of students known to be infected. We provide a simple model for the frequency and size of clusters in this data, based on random arrivals of index cases at schools who then infect their classmates with a highly variable rate, fitting the overdispersion evident in the data. We fit our model to reports from four Canadian provinces, providing estimates of mean and dispersion for cluster size, whilst factoring in imperfect ascertainment. Our parameter estimates are robust to variations in ascertainment fraction. We use these estimates in two ways: i) to explore how uneven the distribution of cases is among different clusters in different jurisdictions (that is, what fraction of cases are in the 20% largest clusters), and ii) to determine the distribution of instantaneous transmission rate β among different index cases. We show how these latter distributions can be used in simulations of school transmission where we explore the effect of different interventions, in the context of highly variable transmission rates.

In planning for upcoming mass vaccinations against COVID-19, many jurisdictions have proposed using primarily age-based rollout strategies, where the oldest are vaccinated first and the youngest last. In the wake of growing evidence that approved vaccines are effective at preventing not only adverse outcomes, but also infection (and hence transmission of SARS-CoV-2), we propose that such age-based rollouts are both less equitable and less effective than strategies that prioritize essential workers. We demonstrate that strategies that target essential workers earlier consistently outperform those that do not, and that prioritizing essential work-ers provides a significant level of indirect protection for older adults. This conclusion holds across numerous outcomes, including cases, hospitalizations, Long COVID, deaths and net monetary benefit, and over a range of possible values for the efficacy of vaccination against infection. Our analysis focuses on regimes where the pandemic continues to be controlled with distancing and other measures as vaccination proceeds, and where the vaccination strategy is expected to last for over the coming 6-8 months — for example British Columbia, Canada. In such a setting with a total population of 5M, vaccinating essential workers sooner is expected to prevent over 200,000 infections, over 600 deaths, and to produce a net monetary benefit of over $500M.

Widespread school closures occurred during the COVID-19 pandemic. Because closures are costly and damaging, many jurisdictions have since reopened schools with control measures in place. Early evidence indicated that schools were low risk and children were unlikely to be very infectious, but it is becoming clear that children and youth can acquire and transmit COVID-19 in school settings and that transmission clusters and outbreaks can be large. We describe the contrasting literature on school transmission, and argue that the apparent discrepancy can be reconciled by heterogeneity, or “overdispersion” in transmission, with many exposures yielding little to no risk of onward transmission, but some unfortunate exposures causing sizeable onward transmission. In addition, respiratory viral loads are as high in children and youth as in adults, pre- and asymptomatic transmission occur, and the possibility of aerosol transmission has been established. We use a stochastic individual-based model to find the implications of these combined observations for cluster sizes and control measures. We consider both individual and environment/activity contributions to the transmission rate, as both are known to contribute to variability in transmission. We find that even small heterogeneities in these contributions result in highly variable transmission cluster sizes in the classroom setting, with clusters ranging from 1 to 20 individuals in a class of 25. None of the mitigation protocols we modeled, initiated by a positive test in a symptomatic individual, are able to prevent large transmission clusters unless the transmission rate is low (in which case large clusters do not occur in any case). Among the measures we modeled, only rapid universal monitoring (for example by regular, onsite, pooled testing) accomplished this prevention. We suggest approaches and the rationale for mitigating these “unfortunate events”, even if they are expected to be rare.

This study aims to characterize distinctive acoustic features of Mandarin tones based on a corpus of 1025 monosyllabic words produced by 21 native Mandarin speakers. For each tone, 22 acoustic cues were extracted. Besides standard F0, duration, and intensity measures, further cues were determined by fitting two mathematical functions to the pitch contours. The first function is a parabola, which gives three parameters: a mean F0, an F0 slope, and an F0 second derivative. The second is a broken-line function, which models the contour as a continuous curve consisting of two lines with a single breakpoint. Cohen's d, sparse Principal Component Analysis, and other statistical measures are used to identify which of the cues, and which combinations of the cues, are important for distinguishing each tone from each other among all the speakers. Although the specific cues that best characterize the tone contours depend on the particular tone and the statistical measure used, this paper shows that the three cues obtained by fitting a parabola to the tone contour are broadly effective. This research suggests using these three cues as a canonical choice for defining tone characteristics.

Exemplar models are a popular class of models used to describe language change. Here we study how limiting the memory capacity of an individual in these models affects the system's behaviour. In particular, we demonstrate the effect this change has on the extinction of categories. Previous work in exemplar dynamics has not addressed this question. In order to investigate this, we will inspect a simplified exemplar model. We will prove for the simplified model that all the sound categories but one will always become extinct, whether memory storage is limited or not. However, computer simulations show that changing the number of stored memories alters how fast categories become extinct.

The Urysohn space is a separable complete metric space with two fundamental properties: (a) universality: every separable metric space can be isometrically embedded in it; (b) ultrahomogeneity: every finite isometry between two finite subspaces can be extended to an auto-isometry of the whole space. The Urysohn space is uniquely determined up to isometry within separable metric spaces by these two properties. We introduce an analogue of the Urysohn space for diversities, a recently developed variant of the concept of a metric space. In a diversity any finite set of points is assigned a non-negative value, extending the notion of a metric which only applies to unordered pairs of points. We construct the unique separable complete diversity that it is ultrahomogeneous and universal with respect to separable diversities., Comment: 15 pages

Transposition is a tendency for organisms to generalize relationships between stimuli in situations where training does not objectively reward relationships over absolute, static associations. Transposition has most commonly been explained as either conceptual understanding of relationships (Köhler, 1938) as nonconceptual effects of neural memory gradients (as in Spence's stimulus discrimination theory, 1937 ). Most behavioral evidence can be explained by the gradient account, but a key finding unexplained by gradients is intermediate transposition, where a central (of three) stimulus, “relationally correct response,” is generalized from training to test. Here, we introduce a dynamic neural field (DNF) model that captures intermediate transposition effects while using neural mechanisms closely resembling those of Spence's original proposal. The DNF model operates on dynamic rather than linear neural relationships, but it still functions by way of gradient interactions, and it does not invoke relational conceptual understanding in order to explain transposition behaviors. In addition to intermediate transposition, the DNF model also replicates the predictions of stimulus discrimination theory with respect to basic two-stimulus transposition. Effects of wider test item spacing were additionally captured. Overall, the DNF model captures a wider range of effects in transposition than stimulus discrimination theory, uses more fully specified neural mechanics, and integrates transposition into a wider modeling effort across cognitive tasks and phenomena. At the same time, the model features a similar low-level focus and emphasis on gradient interactions as Spence's, serving as a conceptual continuation and updating of Spence's work in the field of transposition.

Speech errors involving manipulations of sounds tend to be phonologically regular in the sense that they obey the phonotactic rules of well-formed words. We review the empirical evidence for phonological regularity in prior research, including both categorical assessments of words and regularity at the granular level involving specific segments and contexts. Since the reporting of regularity is affected by human perceptual biases, we also document this regularity in a new data set of 2,228 sublexical errors that was collected using methods that are demonstrably less prone to bias. These facts validate the claim that sound errors are overwhelmingly regular, but the new evidence suggests speech errors admit more phonologically ill-formed words than previously thought. Detailed facts of the phonological structure of errors, including this revised standard, are then related to model assumptions in contemporary theories of phonological encoding. This article is categorized under: Linguistics > Linguistic Theory Linguistics > Computational Models of Language Psychology > Language.

We develop a model of phonological contrast in natural language. Specifically, our model describes the maintenance of contrast between different words in a language, and the elimination of such contrast when sounds in the words merge. An example of such a contrast is that provided by the two vowel sounds i and e, which distinguish pairs of words such as pin and pen in most dialects of English. We model language users' knowledge of the pronunciation of a word as consisting of collections of labeled exemplars stored in memory. Each exemplar is a detailed memory of a particular utterance of the word in question. In our model an exemplar is represented by one or two phonetic variables along with a weight indicating how strong the memory of the utterance is. Starting from an exemplar-level model we derive integro-differential equations for the evolution of exemplar density fields in phonetic space. Using these latter equations we investigate under what conditions two sounds merge, thus eliminating the contrast...

In spoken languages, speakers divide up the space of phonetic possibilities into different regions, corresponding to different phonemes. We consider a simple exemplar model of how this division of phonetic space varies over time among a population of language users. In the particular model we consider, we show that, once the system is initialized with a given set of phonemes, that phonemes do not become extinct: all phonemes will be maintained in the system for all time. This is in contrast to what is observed in more complex models. Furthermore, we show that the boundaries between phonemes fluctuate and we quantitatively study the fluctuations in a simple instance of our model. These results prepare the ground for more sophisticated models in which some phonemes go extinct or new phonemes emerge through other processes., 19 pages

A significant problem in computational language learning is that of inferring the content of well-formedness constraints from input data. In this article, we approach the constraint induction problem as the gradual adjustment of subsymbolic constraints in a connectionist network. In particular, we develop a multi-layer feed-forward network that learns the constraints that underlie restrictions against homorganic consonants, or ‘OCP-Place constraints’, in Arabic roots. The network is trained using standard learning procedures in connection science with a representative sample of Arabic roots. The trained network is shown to classify actual and novel Arabic roots in ways that are qualitatively parallel to a psycholinguistic study of Arabic. Statistical analysis of network behavior also shows that activations of nodes in the hidden layer correspond well with violations of symbolic well-formedness constraints familiar from generative phonology. In sum, it is shown that at least some constraints operative in phonotactic grammar can be learned from data and do not have to be stipulated in advance of learning.

The tight span, or injective envelope, is an elegant and useful construction that takes a metric space and returns the smallest hyperconvex space into which it can be embedded. The concept has stimulated a large body of theory and has applications to metric classification and data visualisation. Here we introduce a generalisation of metrics, called diversities, and demonstrate that the rich theory associated to metric tight spans and hyperconvexity extends to a seemingly richer theory of diversity tight spans and hyperconvexity., Comment: revised in response to referee comments

Diversities are like metric spaces, except that every finite subset, instead of just every pair of points, is assigned a value. Just as there is a theory of minimal distortion embeddings of finite metric spaces into $L_1$, there is a similar, yet undeveloped, theory for embedding finite diversities into the diversity analogue of $L_1$ spaces. In the metric case, it is well known that an $n$-point metric space can be embedded into $L_1$ with $\mathcal{O}(\log n)$ distortion. For diversities, the optimal distortion is unknown. Here, we establish the surprising result that symmetric diversities, those in which the diversity (value) assigned to a set depends only on its cardinality, can be embedded in $L_1$ with constant distortion., 14 pages, 3 figures

A challenge to both understanding and modeling biochemical networks is integrating the effects of diffusion and stochasticity. Here, we use the theory of branching processes to give exact analytical expressions for the mean and variance of protein numbers as a function of time and position in a spatial version of an established model of gene expression. We show that both the mean and the magnitude of fluctuations are determined by the protein’s Kuramoto length—the typical distance a protein diffuses over its lifetime—and find that the covariance between local concentrations of proteins often increases if there are substantial bursts of synthesis during translation. Using high-throughput data, we estimate that the Kuramoto length of cytoplasmic proteins in budding yeast to be an order of magnitude larger than the cell diameter, implying that many such proteins should have an approximately uniform concentration. For constitutively expressed proteins that live substantially longer than their mRNA, we give an exact expression for the deviation of their local fluctuations from Poisson fluctuations. If the Kuramoto length of mRNA is sufficiently small, we predict that such local fluctuations become approximately Poisson in bacteria in much of the cell, unless translational bursting is exceptionally strong. Our results therefore demonstrate that diffusion can act to both increase and decrease the complexity of fluctuations in biochemical networks.

Speech communication can adopt different styles as a function of speaking environments and communicative needs. In auditorily or visually challenging contexts, speakers often alter their speech production using a clarified, hyper-articulated speech style with the intention of enhancing speech intelligibility. Such modifications may result in perceptible articulatory and acoustic changes. Questions thus arise as to whether and what clear-speech modifications facilitate perception. This presentation surveys recent research conducted in our labs, investigating clear-speech production and its associated effects on perception. In a series of three-stream studies, this research relates analyses of visible articulatory features using computer image-processing techniques, measurements of acoustic properties, and perceptual patterns of clear-speech segments and suprasegmentals by native and non-native perceivers. Results reveal that clear (relative to plain) speech modulates different and compensatory articulatory-acoustic cues within each sound category to enhance intelligibility. However, our results also show that clear-speech modifications that reduce phonemic category distinctiveness inhibit intelligibility. These findings indicate that clear-speech effects are governed by the collateral principles of within-category cue enhancement and maintenance of category distinctiveness.

Using mathematical modeling, this study aims to characterize distinctive acoustic features of Mandarin tones based on a corpus of 1013 monosyllabic words produced by 21 native Mandarin speakers. For each tone, 22 acoustic cues were extracted. Besides standard F0, duration, and intensity measures, further cues were determined by fitting two mathematical models to the pitch contours. The first is a broken-line model, which models the contour as a continuous curve consisting of two lines with a single breakpoint. The second model is a parabola, which gives three parameters: a mean F0, an F0 slope, and an F0 curvature. Using Cohen’s d, we identify which of the 22 cues are important for distinguishing each tone from the others for all speakers, as well as identifying cues that are used idiosyncratically by particular speakers. Although the specific cues that best characterize each tone differ, we show that the three cues obtained by fitting a parabola to the tone contour are an effective small set of cues such that any pair of tones is well distinguished by at least one of them. We propose using these three cues as a canonical choice for defining tone characteristics.Using mathematical modeling, this study aims to characterize distinctive acoustic features of Mandarin tones based on a corpus of 1013 monosyllabic words produced by 21 native Mandarin speakers. For each tone, 22 acoustic cues were extracted. Besides standard F0, duration, and intensity measures, further cues were determined by fitting two mathematical models to the pitch contours. The first is a broken-line model, which models the contour as a continuous curve consisting of two lines with a single breakpoint. The second model is a parabola, which gives three parameters: a mean F0, an F0 slope, and an F0 curvature. Using Cohen’s d, we identify which of the 22 cues are important for distinguishing each tone from the others for all speakers, as well as identifying cues that are used idiosyncratically by particular speakers. Although the specific cues that best characterize each tone differ, we show that the three cues obtained by fitting a parabola to the tone contour are an effective small set of cues such...

We have implemented the Jacobian-free Newton-Krylov (JFNK) method to solve the sea ice momentum equation with a viscous-plastic (VP) formulation. The JFNK method has many advantages: the system matrix (the Jacobian) does not need to be formed and stored, the method is parallelizable and the convergence can be nearly quadratic in the vicinity of the solution. The convergence rate of our JFNK implementation is characterized by two phases: an initial phase with slow convergence and a fast phase for which the residual norm decreases significantly from one Newton iteration to the next. Because of this fast phase, the computational gain of the JFNK method over the standard solver used in existing VP models increases with the required drop in the residual norm (termination criterion). The JFNK method is between 3 and 6.6 times faster (depending on the spatial resolution and termination criterion) than the standard solver using a preconditioned generalized minimum residual method. Resolutions tested in this study are 80, 40, 20 and 10km. For a large required drop in the residual norm, both JFNK and standard solvers sometimes do not converge. The failure rate for both solvers increases as the grid is refined but stays relatively small (less than 2.3% of failures). With increasing spatial resolution, the velocity gradients (sea ice deformations) get more and more important. Nonlinear solvers such as the JFNK method tend to have difficulties when there are such sharp structures in the solution. This lack of robustness of both solvers is however a debatable problem as it mostly occurs for large required drops in the residual norm. Furthermore, when it occurs, it usually affects only a few grid cells, i.e., the residual is small for all the velocity components except in very localized regions. Globalization approaches for the JFNK solver, such as the line search method, have not yet proven to be successful. Further investigation is needed.

Molecular dynamics refers to the computer simulation of a material at the atomic level. An open problem in numerical analysis is to explain the apparent reliability of molecular dynamics simulations. The difficulty is that individual trajectories computed in molecular dynamics are accurate for only short time intervals, whereas apparently reliable information can be extracted from very long-time simulations. It has been conjectured that long molecular dynamics trajectories have low-dimensional statistical features that accurately approximate those of the original system. Another conjecture is that numerical trajectories satisfy the shadowing property: that they are close over long time intervals to exact trajectories but with different initial conditions. We prove that these two views are actually equivalent to each other, after we suitably modify the concept of shadowing. A key ingredient of our result is a general theorem that allows us to take random elements of a metric space that are close in distribution and embed them in the same probability space so that they are close in a strong sense. This result is similar to the Strassen-Dudley Theorem except that a mapping is provided between the two random elements. Our results on shadowing are motivated by molecular dynamics but apply to the approximation of any dynamical system when initial conditions are selected according to a probability measure., Comment: 21 pages, final version accepted in SIAM Dyn Sys

We present the results of a set of numerical experiments designed to investigate the appropriateness of various integration schemes for molecular dynamics simulations. In particular, we wish to identify which numerical methods, when applied to an ergodic Hamiltonian system, sample the state-space in an unbiased manner. We do this by describing two Hamiltonian system for which we can analytically compute some of the important statistical features of its trajectories, and then applying various numerical integration schemes to them. We can then compare the results from the numerical simulation against the exact results for the system and see how closely they agree. The statistic we study is the empirical distribution of particle velocity over long trajectories of the systems. We apply four methods: one symplectic method (Störmer–Verlet) and three energy-conserving step-and-project methods. The symplectic method performs better on both test problems, accurately computing empirical distributions for all step-lengths consistent with stability. Depending on the test system and the method, the step-and-project methods are either no longer ergodic for any step length (thus giving the wrong empirical distribution) or give the correct distribution only in the limit of step-size going to zero.

We derive a test problem for evaluating the ability of time-stepping methods to preserve statistical properties of systems in molecular dynamics. We consider a family of deterministic systems consisting of a finite number of particles interacting on a compact interval. The particles are given random initial conditions and interact through instantaneous energy- and momentum-conserving collisions. As the number of particles, the particle density, and the mean particle speed go to infinity, the trajectory of a tracer particle is shown to converge to a stationary Gaussian stochastic process. We approximate this system by one described by a system of ordinary differential equations and provide numerical evidence that it converges to the same stochastic process. We simulate the latter system with a variety of numerical integrators, including the symplectic Euler method, a fourth-order Runge-Kutta method, and an energy-conserving step-and-project method. We assess the methods’ ability to recapture the system’s limiting statistics and observe that symplectic Euler performs significantly better than the others for comparable computational expense.

We discuss the long-time numerical simulation of Hamiltonian systems of ordinary differential equations. Our goal is to explain the ability of symplectic integration schemes such as the Stormer--Verlet method to compute accurate long-time averages for these systems in the context of molecular dynamics. This paper introduces a weakened version of ergodicity that allows us to study this problem. First, we demonstrate the utility of the weakened ergodicity definition by showing that it is a property of Hamiltonian systems robust to perturbations. Second, we study what the weakened ergodicity of a Hamiltonian system implies about numerical simulations of the system. In the case where a numerical method is volume-conserving and approximately energy-conserving, we show that long-time averages are approximated well for sufficiently small step lengths.

We study the long-time behaviour of large systems of ordinary differential equations with random data. Our main focus is a Hamiltonian system which describes a distinguished particle attached to a large collection of heat bath particles by springs. In the limit where the size of the heat bath tends to infinity, the trajectory of the distinguished particle can be weakly approximated, on finite time intervals, by a Langevin stochastic differential equation. We examine the long-term behaviour of these trajectories, both analytically and numerically. We find ergodic behaviour manifest in both the long-time empirical measures and in the resulting auto-correlation functions.

Public health researchers and practitioners have begun to recognize the dynamic nature of disability, promote the health of people with disabilities, and develop strategies to prevent secondary conditions among them. To understand the epidemiology of secondary conditions, the authors developed the Massachusetts Survey of Secondary Conditions, a longitudinal study of adults with major disabilities (n = 656) based on a conceptual framework linking disability, mediating factors, and health outcomes. This paper reports baseline data on the number of secondary conditions experienced by survey respondents. Respondents experienced a mean of 5.3 of 17 secondary conditions. More numerous secondary conditions were associated with fair or poor general health and number of days unable to do routine activities. Factors amenable to public health interventions included difficulty with weight and exercise maintenance, tobacco and marijuana use, and experiencing assault. Disability should be a focus in all public health research, policy, and programs.

An adaptive model reduction algorithm is proposed for systems of ODEs from chemical kinetics. Its goal is to provide an accurate approximation to the solution of these systems faster than could be obtained through straightforward numerical integration. The algorithm approximates a system with a sequence of reduced models, each one appropriate to the dynamics of the system during a period of the trajectory. Reduced models are identical to the original system except for the deletion of some chemical reactions. This saves the cost of computing unimportant reaction coefficients. Both the reduced models and the durations for which they are used are selected adaptively in order to efficiently yield an accurate approximate solution. The performance of the algorithm is assessed through numerical experiments.

Special issu PRIMA 2013, The embedding of finite metrics in 1 has become a fundamental tool for both combinatorial optimization and large-scale data analysis. One important application is to network flow problems as there is close relation between max-flow min-cut theorems and the minimal distortion embeddings of metrics into 1. Here we show that this theory can be generalized to a larger set of combinatorial optimization problems on both graphs and hypergraphs. This theory is not built on metrics and metric embeddings, but on diversities, a type of multi-way metric introduced recently by the authors. We explore diversity embeddings, 1 diversities, and their application to Steiner Tree Packing and Hypergraph Cut problems.

Consider a particle diffusing in a confined volume which is divided into two equal regions. In one region the diffusion coefficient is twice the value of the diffusion coefficient in the other region. Will the particle spend equal proportions of time in the two regions in the long term? Statistical mechanics would suggest yes, since the number of accessible states in each region is presumably the same. However, another line of reasoning suggests that the particle should spend less time in the region with faster diffusion, since it will exit that region more quickly. We demonstrate with a simple microscopic model system that both predictions are consistent with the information given. Thus, specifying the diffusion rate as a function of position is not enough to characterize the behaviour of a system, even assuming the absence of external forces. We propose an alternative framework for modelling diffusive dynamics in which both the diffusion rate and equilibrium probability density for the position of the particle are specified by the modeller. We introduce a numerical method for simulating dynamics in our framework that samples from the equilibrium probability density exactly and is suitable for discontinuous diffusion coefficients., 21 pages, 6 figures. Second round of revisions. This is the version that will appear in Proc Roy Soc

White noise-driven nonlinear stochastic partial differential equations (SPDEs) of parabolic type are frequently used to model physical and biological systems in space dimensions d = 1,2,3. Whereas existence and uniqueness of weak solutions to these equations are well established in one dimension, the situation is different for d \geq 2. Despite their popularity in the applied sciences, higher dimensional versions of these SPDE models are generally assumed to be ill-posed by the mathematics community. We study this discrepancy on the specific example of the two dimensional Allen-Cahn equation driven by additive white noise. Since it is unclear how to define the notion of a weak solution to this equation, we regularize the noise and introduce a family of approximations. Based on heuristic arguments and numerical experiments, we conjecture that these approximations exhibit divergent behavior in the continuum limit. The results strongly suggest that a series of published numerical studies are problematic: shrinking the mesh size in these simulations does not lead to the recovery of a physically meaningful limit., 21 pages, 4 figures; accepted by Journal of Computational Physics (Dec 2011)

An open problem in numerical analysis is to explain why molecular dynamics works. The difficulty is that numerical trajectories are only accurate for very short times, whereas the simulations are performed over long time intervals. It is believed that statistical information from these simulations is accurate, but no one has offered a rigourous proof of this. In order to give mathematicians a clear goal in understanding this problem, we state a precise mathematical conjecture about molecular dynamics simulation of a particular system. We believe that if the conjecture is proved, we will then understand why molecular dynamics works.

Phase field crystals (PFC) are a tool for simulating materials at the atomic level. They combine the small length-scale resolution of molecular dynamics (MD) with the ability to simulate dynamics on mesoscopic time scales. We show how PFC can be interpreted as the result of applying coarse-graining in time to the microscopic density field of molecular dynamics simulations. We take the form of the free energy for the phase field from the classical density functional theory of inhomogeneous liquids and then choose coefficients to match the structure factor of the time coarse-grained microscopic density field. As an example, we show how to construct a PFC free energy for Weber and Stillinger's two-dimensional square crystal potential which models a system of proteins suspended in a membrane., Comment: 5 pages, 4 figures, typos corrected, more explanation in parts, equilib vs non-equilib clarified

We consider the weak convergence of numerical methods for stochastic differential equations (SDEs). Weak convergence is usually expressed in terms of the convergence of expected values of test functions of the trajectories. Here we present an alternative formulation of weak convergence in terms of the well-known Prokhorov metric on spaces of random variables. For a general class of methods, we establish bounds on the rates of convergence in terms of the Prokhorov metric. In doing so, we revisit the original proofs of weak convergence and show explicitly how the bounds on the error depend on the smoothness of the test functions. As an application of our result, we use the Strassen - Dudley theorem to show that the numerical approximation and the true solution to the system of SDEs can be re-embedded in a probability space in such a way that the method converges there in a strong sense. One corollary of this last result is that the method converges in the Wasserstein distance, another metric on spaces of random variables. Another corollary establishes rates of convergence for expected values of test functions assuming only local Lipschitz continuity. We conclude with a review of the existing results for pathwise convergence of weakly converging methods and the corresponding strong results available under re-embedding., Comment: 12 pages, 2nd revision for IMA J Numerical Analysis. Further minor errors corrected

We propose a continuous model for variation in the evolutionary rate across sites and over the phylogenetic tree. We derive exact transition probabilities of substitutions under this model. Changes in rate are modelled using the CIR process, a diffusion widely used in financial applications. The model directly extends the standard gamma distributed rates across site model, with one additional parameter governing changes in rate down the tree. The parameters of the model can be estimated directly from two well-known statistics: the index of dispersion and the gamma shape parameter of the rates across sites model. The CIR model can be readily incorporated into probabilistic models for sequence evolution. We provide here an exact formula for the likelihood of a three-taxon tree. The likelihoods of larger trees can be evaluated using Monte–Carlo methods.