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The Jacobi-Maupertuis metric provides a reformulation of the classical N-bodyproblem as a geodesic flow on an energy-dependent metric space denoted $M_E$where $E$ is the energy of the problem. We show that $M_E$ has finite diameterfor $E < 0$. Consequently $M_E$ has no metric rays. Motivation comes from workof Burgos- Maderna and Polimeni-Terracini for the case $E \ge 0$ and from aneed to correct an error made in a previous ``proof''. We show that $M_E$ hasfinite diameter for $E < 0$ by showing that there is a constant $D$ such thatall points of the Hill region lie a distance $D$ from the Hill boundary. (When$E \ge 0$ the Hill boundary is empty.) The proof relies on a game of escapewhich allows us to quantify the escape rate from a closed subset ofconfiguration space, and the reduction of this game to one of escaping theboundary of a polyhedral convex cone into its interior.

The Kepler problem is the special case $\alpha = 1$ of the power law problem:to solve Newton's equations for a central force whose potential is of the form$-\mu/r^{\alpha}$ where $\mu$ is a coupling constant. Associated to such aproblem is a two-dimensional cone with cone angle $2 \pi c$ with $c = 1 -\frac{\alpha}{2}$. We construct a transformation taking the geodesics of thiscone to the zero energy solutions of the $\alpha$-power law problem. The`Kepler Cone' is the cone associated to the Kepler problem. This zero-energycone transformation is a special case of a transformation discovered byMaclaurin in the 1740s transforming the $\alpha$- power law problem for anyenergies to a `Maclaurin dual' $\gamma$-power law problem where $\gamma =\frac{2 \alpha}{2-\alpha}$ and which, in the process, mixes up the energy ofone problem with the coupling constant of the other. We derive Maclaurinduality using the Jacobi-Maupertuis metric reformulation of mechanics. We thenuse the conical metric to explain properties of Rutherford-type scattering offpower law potentials at positive energies. The one possibly new result in thepaper concerns ``star-burst curves'' which arise as limits of families negativeenergy solutions as their angular momentum tends to zero. We relate geodesicscattering on the cone to Rutherford type scattering of beams of solutions inthe potential. We describe some history around Maclaurin duality and give twoderivations of the Jacobi-Maupertuis metric reformulation of classicalmechanics. The piece is expository, aimed at an upper-division undergraduate.Think American Math. Monthly.

For n bodies moving in Euclidean d-space under the influence of a homogeneouspair interaction we compactify every center-of-mass energy surface, obtaining a2d(n -1)-1 - dimensional manifold with corners in the sense of Melrose. After atime change, the flow on this manifold is globally defined and non-trivial onthe boundary.

This file is composed of questions that emerged or were of interest duringthe workshop "Interactions between Descriptive Set Theory and Smooth Dynamics"that took place in Banff, Canada on 2022.

The infinite spin problem concerns the rotational behavior of total collisionorbits in the $n$-body problem. It has long been known that when a solutiontends to total collision then its normalized configuration curve must convergeto the set of normalized central configurations. In the planar n-body problemevery normalized configuration determines a circle of rotationally equivalentnormalized configurations and, in particular, there are circles of normalizedcentral configurations. It's conceivable that by means of an infinite spin, atotal collision solution could converge to such a circle instead of to aparticular point on it. Here we prove that this is not possible, at least ifthe limiting circle of central configurations is isolated from other circles ofcentral configurations. (It is believed that all central configurations areisolated, but this is not known in general.) Our proof relies on combining thecenter manifold theorem with the Lojasiewicz gradient inequality.

The subject is brake orbits for the 3-body problem: orbits where allvelocities are zero at some instant. We extract a paradox and a mystery out ofa recent database of 30 non-collision periodic brake orbits for the equal mass3-body problem built up by Li and Liao. We solve the paradox and the mysteryand pose an open question.

The paper considers the equivalence relation of conjugacy-by-homeomorphism ondiffeomorphisms of smooth manifolds. In dimension 2 and above it is shown thatthere is no Borel method of attaching complete numerical invariants. Indimension 5 and above it is shown that the equivalence relation is not Borel,and in fact is complete analytic.

Although projection-based reduced-order models (ROMs) for parameterizednonlinear dynamical systems have demonstrated exciting results across a rangeof applications, their broad adoption has been limited by their intrusivity:implementing such a reduced-order model typically requires significantmodifications to the underlying simulation code. To address this, we propose amethod that enables traditionally intrusive reduced-order models to beaccurately approximated in a non-intrusive manner. Specifically, the approachapproximates the low-dimensional operators associated with projection-basedreduced-order models (ROMs) using modern machine-learning regressiontechniques. The only requirement of the simulation code is the ability toexport the velocity given the state and parameters as this functionality isused to train the approximated low-dimensional operators. In addition toenabling nonintrusivity, we demonstrate that the approach also leads to verylow computational complexity, achieving up to $1000\times$ reduction in runtime. We demonstrate the effectiveness of the proposed technique on two typesof PDEs.

In quantum mechanics, states are described by density matrices. Though their probabilistic interpretation is rooted in ensemble theory, density matrices embody a known shortcoming. They do not completely express the physical realization of an ensemble. Conveniently, the outcome statistics of projective and positive operator-valued measurements do not depend on the ensemble realization, only on the density matrix. Here, we show how the geometric approach to quantum mechanics tracks ensemble realizations. We do so in two concrete cases of a finite-dimensional quantum system interacting with another one with (i) finite-dimensional Hilbert space, relevant for quantum thermodynamics, and (ii) infinite-dimensional Hilbert space, relevant for state-manipulation protocols.

We show that non-obtuse trapezoids are uniquely determined by their Dirichlet Laplace spectrum. This extends our previous result [Hezari et al., Ann. Henri Poincare 18(12), 3759-3792 (2017)], which was only concerned with the Neumann Laplace spectrum.

Hidden Markov chains are widely applied statistical models of stochastic processes, from fundamental physics and chemistry to finance, health, and artificial intelligence. The hidden Markov processes they generate are notoriously complicated, however, even if the chain is finite state: no finite expression for their Shannon entropy rate exists, as the set of their predictive features is generically infinite. As such, to date one cannot make general statements about how random they are nor how structured. Here, we address the first part of this challenge by showing how to efficiently and accurately calculate their entropy rates. We also show how this method gives the minimal set of infinite predictive features. A sequel addresses the challenge’s second part on structure.

We construct multidimensional Schrödinger operators with a spectrum that has no gaps at high energies and that is nowhere dense at low energies. This gives the first example for which this widely expected topological structure of the spectrum in the class of uniformly recurrent Schrödinger operators, namely the coexistence of a half-line and a Cantor-type structure, can be confirmed. Our construction uses Schrödinger operators with separable potentials that decompose into one-dimensional potentials generated by the Fibonacci sequence and relies on the study of such operators via the trace map and the Fricke-Vogt invariant. To show that the spectrum contains a half-line, we prove an abstract Bethe–Sommerfeld criterion for sums of Cantor sets which may be of independent interest.

Nonequilibrium information thermodynamics determines the minimum energy dissipation to reliably erase memory under time-symmetric control protocols. We demonstrate that its bounds are tight and so show that the costs overwhelm those implied by Landauer’s energy bound on information erasure. Moreover, in the limit of perfect computation, the costs diverge. The conclusion is that time-asymmetric protocols should be developed for efficient, accurate thermodynamic computing. And, that Landauer’s Stack—the full suite of theoretically-predicted thermodynamic costs—is ready for experimental test and calibration.

Fundamental to many transportation network studies, traffic flow models can be used to describe traffic dynamics determined by drivers' car-following, lane-changing, merging, and diverging behaviors. In this study, we develop a deterministic queueing model of network traffic flow, in which traffic on each link is considered as a queue. In the link queue model (LQM), the demand and supply of a link queue are defined in the queue size (number of vehicles), and its in- and out-flows are computed from junction flux functions corresponding to macroscopic merging and diverging rules. The new model is a system of ordinary differential equations that is mathematically tractable and computationally efficient and can capture queue spillbacks and interactions among links. We further demonstrate that the LQM is fundamentally different from the cell transmission model (CTM) and link transmission model (LTM) for a road segment, a signalized ring road, and a diverge-merge network, with respect to the shock and rarefaction waves, network fundamental diagram, and stability property. In a sense, the new model is a spacecontinuous approximation of the kinematic wave model and can be a useful addition to the multiscale modeling framework of network traffic flow. The model has been applied to formulate and solve network traffic control and observation problems.

Even simply-defined, finite-state generators produce stochastic processesthat require tracking an uncountable infinity of probabilistic features foroptimal prediction. For processes generated by hidden Markov chains theconsequences are dramatic. Their predictive models are genericallyinfinite-state. And, until recently, one could determine neither theirintrinsic randomness nor structural complexity. The prequel, though, introducedmethods to accurately calculate the Shannon entropy rate (randomness) and toconstructively determine their minimal (though, infinite) set of predictivefeatures. Leveraging this, we address the complementary challenge ofdetermining how structured hidden Markov processes are by calculating theirstatistical complexity dimension -- the information dimension of the minimalset of predictive features. This tracks the divergence rate of the minimalmemory resources required to optimally predict a broad class of truly complexprocesses.

We consider random products of SL(2,R) matrices that depend on a parameter in a non-uniformly hyperbolic regime. We show that if the dependence on the parameter is monotone then almost surely the random product has upper (limsup) Lyapunov exponent that is equal to the value prescribed by the Furstenberg Theorem (and hence positive) for all parameters, but the lower (liminf) Lyapunov exponent is equal to zero for a dense Gδ set of parameters of zero Hausdorff dimension. As a byproduct of our methods, we provide a purely geometrical proof of Spectral Anderson Localization for discrete Schrödinger operators with random potentials (including the Anderson-Bernoulli model) on a one dimensional lattice.

We consider random products of SL(2,R) matrices that depend on a parameter in a non-uniformly hyperbolic regime. We show that if the dependence on the parameter is monotone then almost surely the random product has upper (limsup) Lyapunov exponent that is equal to the value prescribed by the Furstenberg Theorem (and hence positive) for all parameters, but the lower (liminf) Lyapunov exponent is equal to zero for a dense Gδ set of parameters of zero Hausdorff dimension. As a byproduct of our methods, we provide a purely geometrical proof of Spectral Anderson Localization for discrete Schrödinger operators with random potentials (including the Anderson-Bernoulli model) on a one dimensional lattice.

Oscillator networks display intricate synchronization patterns. Determiningtheir stability typically requires incorporating the symmetries of the networkcoupling. Going beyond analyses that appeal only to a network's automorphismgroup, we explore synchronization patterns that emerge from the phase-shiftinvariance of the dynamical equations and symmetries in the nodes. We show thatthese nonstructural symmetries simplify stability calculations. We analyze aring-network of phase-amplitude oscillators that exhibits a "decoupled" statein which physically-coupled nodes appear to act independently due to emergentcancellations in the equations of dynamical evolution. We establish that thisstate can be linearly stable for a ring of phase-amplitude oscillators, but notfor a ring of phase-only oscillators that otherwise require explicitlong-range, nonpairwise, or nonphase coupling. In short, amplitude-phaseinteractions are key to stable synchronization at a distance.

All computation is physically embedded. Reflecting this, a growing body ofresults embraces rate equations as the underlying mechanics of thermodynamiccomputation and biological information processing. Strictly applying theimplied continuous-time Markov chains, however, excludes a universe of naturalcomputing. We show that expanding the toolset to continuous-time hidden Markovchains substantially removes the constraints. The general point is madeconcrete by our analyzing two eminently-useful computations that are impossibleto describe with a set of rate equations over the memory states. We design andanalyze a thermodynamically-costless bit flip, providing a first counterexampleto rate-equation modeling. We generalize this to a costless Fredkin gate---akey operation in reversible computing that is computation universal. Goingbeyond rate-equation dynamics is not only possible, but necessary if stochasticthermodynamics is to become part of the paradigm for physical informationprocessing.

When an experimentalist measures a time series of qubits, the outcomes constitute a classical stochastic process. We show that projective measurement induces high complexity in these processes in two specific senses: They are inherently random (finite Shannon entropy rate) and they require infinite memory for optimal prediction (divergent statistical complexity). We identify nonorthogonality of the quantum states as the mechanism underlying the resulting complexities and examine the influence that measurement choice has on the randomness and structure of measured qubit processes. We introduce quantitative measures of this complexity and provide efficient algorithms for their estimation.

In this paper, a generalized fractional-order SEIR model is proposed, denoted by SEIQRP model, which divided the population into susceptible, exposed, infectious, quarantined, recovered and insusceptible individuals and has a basic guiding significance for the prediction of the possible outbreak of infectious diseases like the coronavirus disease in 2019 (COVID-19) and other insect diseases in the future. Firstly, some qualitative properties of the model are analyzed. The basic reproduction number R0 is derived. When R01 , the endemic equilibrium point is also unique. Furthermore, some conditions are established to ensure the local asymptotic stability of disease-free and endemic equilibrium points. The trend of COVID-19 spread in the USA is predicted. Considering the influence of the individual behavior and government mitigation measurement, a modified SEIQRP model is proposed, defined as SEIQRPD model, which is divided the population into susceptible, exposed, infectious, quarantined, recovered, insusceptible and dead individuals. According to the real data of the USA, it is found that our improved model has a better prediction ability for the epidemic trend in the next two weeks. Hence, the epidemic trend of the USA in the next two weeks is investigated, and the peak of isolated cases is predicted. The modified SEIQRP model successfully capture the development process of COVID-19, which provides an important reference for understanding the trend of the outbreak.

Maxwellian ratchets are autonomous, finite-state thermodynamic engines that implement input-output informational transformations. Previous studies of these "demons"focused on how they exploit environmental resources to generate work: They randomize ordered inputs, leveraging increased Shannon entropy to transfer energy from a thermal reservoir to a work reservoir while respecting both Liouvillian state-space dynamics and the second law. However, to date, correctly determining such functional thermodynamic operating regimes was restricted to a very few engines for which correlations among their information-bearing degrees of freedom could be calculated exactly and in closed form, a highly restricted set. Additionally, a key second dimension of ratchet behavior was largely ignored: Ratchets do not merely change the randomness of environmental inputs, their operation constructs and deconstructs patterns. To address both dimensions, we adapt recent results from dynamical-systems and ergodic theories that efficiently and accurately calculate the entropy rates and the rate of statistical complexity divergence of general hidden Markov processes. In concert with the information processing second law, these methods accurately determine thermodynamic operating regimes for finite-state Maxwellian demons with arbitrary numbers of states and transitions. In addition, they facilitate analyzing structure versus randomness tradeoffs that a given engine makes. The result is a greatly enhanced perspective on the information processing capabilities of information engines. As an application, we give a thorough-going analysis of the Mandal-Jarzynski ratchet, demonstrating that it has an uncountably infinite effective state space.

Building on parallels between geometric quantum mechanics and classicalmechanics, we explore an alternative basis for quantum thermodynamics thatexploits the differential geometry of the underlying state space. We developboth microcanonical and canonical ensembles, introducing continuous mixedstates as distributions on the manifold of quantum states. We call out theexperimental consequences for a gas of qudits. We define quantum heat and workin an intrinsic way, including single-trajectory work, and reformulatethermodynamic entropy in a way that accords with classical, quantum, andinformation-theoretic entropies. We give both the First and Second Laws ofThermodynamics and Jarzynki's Fluctuation Theorem. The result is a moretransparent physics, than conventionally available, in which the mathematicalstructure and physical intuitions underlying classical and quantum dynamics areseen to be closely aligned.

Density matrices capture all of a quantum system's statistics accessiblethrough projective and positive operator-valued measurements. They do notcompletely determine its state, however, as they neglect the physicalrealization of ensembles. Fortunately, the concept of geometric quantum statedoes properly describe physical ensembles. Here, given knowledge of a densitymatrix, possibly arising from a tomography protocol, we show how to estimatethe geometric quantum state using a maximum entropy principle based on ageometrically-appropriate entropy.

In the end of 2019, a new type of coronavirus first appeared in Wuhan. Through the real-data of COVID-19 from January 23 to March 18, 2020, this paper proposes a fractional SEIHDR model based on the coupling effect of inter-city networks. At the same time, the proposed model considers the mortality rates (exposure, infection and hospitalization) and the infectivity of individuals during the incubation period. By applying the least squares method and prediction-correction method, the proposed system is fitted and predicted based on the real-data from January 23 to March 18 - m where m represents predict days. Compared with the integer system, the non-network fractional model has been verified and can better fit the data of Beijing, Shanghai, Wuhan and Huanggang. Compared with the no-network case, results show that the proposed system with inter-city network may not be able to better describe the spread of disease in China due to the lock and isolation measures, but this may have a significant impact on countries that has no closure measures. Meanwhile, the proposed model is more suitable for the data of Japan, the USA from January 22 and February 1 to April 16 and Italy from February 24 to March 31. Then, the proposed fractional model can also predict the peak of diagnosis. Furthermore, the existence, uniqueness and boundedness of a nonnegative solution are considered in the proposed system. Afterward, the disease-free equilibrium point is locally asymptotically stable when the basic reproduction number R 0 ≤ 1 , which provide a theoretical basis for the future control of COVID-19.

Adaptive systems -- such as a biological organism gaining survival advantage,an autonomous robot executing a functional task, or a motor proteintransporting intracellular nutrients -- must model the regularities andstochasticity in their environments to take full advantage of thermodynamicresources. Analogously, but in a purely computational realm, machine learningalgorithms estimate models to capture predictable structure and identifyirrelevant noise in training data. This happens through optimization ofperformance metrics, such as model likelihood. If physically implemented, isthere a sense in which computational models estimated through machine learningare physically preferred? We introduce the thermodynamic principle that workproduction is the most relevant performance metric for an adaptive physicalagent and compare the results to the maximum-likelihood principle that guidesmachine learning. Within the class of physical agents that most efficientlyharvest energy from their environment, we demonstrate that an efficient agent'smodel explicitly determines its architecture and how much useful work itharvests from the environment. We then show that selecting the maximum-workagent for given environmental data corresponds to finding themaximum-likelihood model. This establishes an equivalence betweennonequilibrium thermodynamics and dynamic learning. In this way, workmaximization emerges as an organizing principle that underlies learning inadaptive thermodynamic systems.

Designing deep neural networks is an art that often involves an expensivesearch over candidate architectures. To overcome this for recurrent neural nets(RNNs), we establish a connection between the hidden state dynamics in an RNNand gradient descent (GD). We then integrate momentum into this framework andpropose a new family of RNNs, called {\em MomentumRNNs}. We theoretically proveand numerically demonstrate that MomentumRNNs alleviate the vanishing gradientissue in training RNNs. We study the momentum long-short term memory(MomentumLSTM) and verify its advantages in convergence speed and accuracy overits LSTM counterpart across a variety of benchmarks. We also demonstrate thatMomentumRNN is applicable to many types of recurrent cells, including those inthe state-of-the-art orthogonal RNNs. Finally, we show that other advancedmomentum-based optimization methods, such as Adam and Nesterov acceleratedgradients with a restart, can be easily incorporated into the MomentumRNNframework for designing new recurrent cells with even better performance. Thecode is available at https://github.com/minhtannguyen/MomentumRNN.

Memcomputing is a novel computing paradigm that employs time non-local dynamical systems to compute with and in memory. The digital version of these machines [digital memcomputing machines or (DMMs)] is scalable, and is particularly suited to solve combinatorial optimization problems. One of its possible realizations is by means of standard electronic circuits, with and without memory. Since these elements are non-quantum, they can be described by ordinary differential equations. Therefore, the circuit representation of DMMs can also be simulated efficiently on our traditional computers. We have indeed previously shown that these simulations only require time and memory resources that scale linearly with the problem size when applied to finding a good approximation to the optimum of hard instances of the maximum-satisfiability problem. The state-of-the-art algorithms, instead, require exponential resources for the same instances. However, in that work, we did not push the simulations to the limit of the processor used. Since linear scalability at smaller problem sizes cannot guarantee linear scalability at much larger sizes, we have extended these results in a stress-test up to 64×106 variables (corresponding to about 1 billion literals), namely the largest case that we could fit on a single core of an Intel Xeon E5-2860 with 128 GB of dynamic random-access memory (DRAM). For this test, we have employed a commercial simulator, Falcon of MemComputing, Inc. We find that the simulations of DMMs still scale linearly in both time and memory up to these very large problem sizes versus the exponential requirements of the state-of-the-art solvers. These results further reinforce the advantages of the physics-based memcomputing approach compared with traditional ones.

Vickrey (1991, 2020) proposed a bathtub model for the evolution of trip flows served by privately operated vehicles inside a road network based on three premises: (i) treatment of the road network as a single bathtub; (ii) the speed-density relation at the network level, also known as the network fundamental diagram of vehicular traffic, and (iii) the time-independent negative exponential distribution of trip distances. However, the distributions of trip distances are generally time-dependent in the real world, and Vickrey's model leads to unreasonable results for other types of trip distance distributions. Thus there is a need to develop a bathtub model with more general trip distance distribution patterns. In this study, we present a unified framework for modeling network trip flows with general distributions of trip distances, including negative exponential, constant, and regularly sorting trip distances studied in the literature. In addition to tracking the number of active trips as in Vickrey's model, this model also tracks the evolution of the distribution of active trips’ remaining distances. We derive four equivalent differential formulations from the network fundamental diagram and the conservation law of trips for the number of active trips with remaining distances not smaller than any value. Then we define and discuss the properties of stationary and gridlock states, derive the integral form of the bathtub model with the characteristic method, and present two numerical methods to solve the bathtub model based on the differential and integral forms respectively. We further study equivalent formulations and solutions for two special types of distributions of trip distances: time-independent negative exponential or deterministic. In particular, we present six equivalent conditions for Vickrey's bathtub model to be applicable.

Starting from measured data, we develop a method to compute the finestructure of the spectrum of the Koopman operator with rigorous convergenceguarantees. The method is based on the observation that, in themeasure-preserving ergodic setting, the moments of the spectral measureassociated to a given observable are computable from a single trajectory ofthis observable. Having finitely many moments available, we use the classicalChristoffel-Darboux kernel to separate the atomic and absolutely continuousparts of the spectrum, supported by convergence guarantees as the number ofmoments tends to infinity. In addition, we propose a technique to detect thesingular continuous part of the spectrum as well as two methods to approximatethe spectral measure with guaranteed convergence in the weak topology,irrespective of whether the singular continuous part is present or not. Theproposed method is simple to implement and readily applicable to large-scalesystems since the computational complexity is dominated by inverting an$N\times N$ Hermitian positive-definite Toeplitz matrix, where $N$ is thenumber of moments, for which efficient and numerically stable algorithms exist;in particular, the complexity of the approach is independent of the dimensionof the underlying state-space. We also show how to compute, from measured data,the spectral projection on a given segment of the unit circle, allowing us toobtain a finite-dimensional approximation of the operator that explicitly takesinto account the point and continuous parts of the spectrum. Finally, wedescribe a relationship between the proposed method and the so-called HankelDynamic Mode Decomposition, providing new insights into the behavior of theeigenvalues of the Hankel DMD operator. A number of numerical examplesillustrate the approach, including a study of the spectrum of the lid-driventwo-dimensional cavity flow.

We study certain dynamical systems which leave invariant an indefinitequadratic form via semigroups or evolution families of complex symmetricHilbert space operators. In the setting of bounded operators we show that a$\mathcal{C}$-selfadjoint operator generates a contraction $C_0$-semigroup ifand only if it is dissipative. In addition, we examine the abstract Cauchyproblem for nonautonomous linear differential equations possessing a complexsymmetry. In the unbounded operator framework we isolate the class of\emph{complex symmetric, unbounded semigroups} and investigate Stone-typetheorems adapted to them. On Fock space realization, we characterize all$\mathcal{C}$-selfadjoint, unbounded weighted composition semigroups. As abyproduct we prove that the generator of a $\mathcal{C}$-selfadjoint, unboundedsemigroup is not necessarily $\mathcal{C}$-selfadjoint.

We consider a certain class of nonlinear maps that preserve the probability simplex, i.e., stochastic maps, which are inspired by the DeGroot-Friedkin model of belief/opinion propagation over influence networks. The corresponding dynamical models describe the evolution of the probability distribution of interacting species. Such models where the probability transition mechanism depends nonlinearly on the current state are often referred to as nonlinear Markov chains. In this paper, we develop stability results and study the behavior of representative opinion models. The stability certificates are based on the contractivity of the nonlinear evolution in the ℓ1-metric. We apply the theory to two types of opinion models where the adaptation of the transition probabilities to the current state is exponential and linear - both of these can display a wide range of behaviors. We discuss continuous-time and other generalizations.

All existing day-to-day dynamics of departure time choice at a single bottleneck are unstable, and this has led to doubt over the existence of a stable user equilibrium in the real world. However, empirical observations and our personal driving experience suggest stable stationary congestion patterns during a peak period. In this paper, we attempt to reconcile the discrepancy by presenting a stable day-to-day dynamical system for drivers' departure time choice at a single bottleneck. In our model, the decision variable in the execution stage is still drivers' departure times on the next day, but in the planning stage before the execution stage, drivers determine their departure times in order to arrive at the destination at better times with lower scheduling costs. We first define within-day traffic dynamics with the point queue model, costs, the departure time user equilibrium (DTUE), and the arrival time user equilibrium (ATUE). We then identify three behavioral principles in the planning stage: (i) drivers choose their departure and arrival times in a backward fashion (backward choice principle); (ii) after choosing the arrival times, they update their departure times to balance the total costs (cost-balancing principle); (iii) they choose their arrival times to reduce their scheduling costs or gain their scheduling payoffs (scheduling cost-reducing or scheduling payoff-gaining principle). In this sense, drivers' departure and arrival time choices are driven by their scheduling payoff choice. With a single tube or imaginary road model, we convert the nonlocal day-to-day arrival time shifting problem to a local scheduling payoff shifting problem. After introducing a new variable for the imaginary density, we apply the Lighthill-Whitham-Richards (LWR) model to describe the day-to-day dynamics of scheduling payoff choice and present splitting and costbalancing schemes to determine arrival and departure flow rates accordingly. We define the scheduling payoff user equilibrium (SPUE) as the stationary state of the LWR model, formulate a new optimization problem for the SPUE, and prove the global stability of the SPUE and, therefore, ATUE and DTUE by using Lyapunov's second method in which the objective function in the optimization formulation is the potential function. We also develop the corresponding discrete models for numerical solutions and use one numerical example to demonstrate the effectiveness and stability of the new day-to-day dynamical model. Different from existing ones, the new adjustment mechanism leads to stable day-to-day departure time choice dynamics by guaranteeing that drivers have better choices of departure/arrival times with larger scheduling payoffs on the next day, and such better choices are not over-chosen because of the constraint imposed by the single tube's cross-section area, which is equal to the jam density in the LWR model. This study is the first step for understanding stable day-to-day dynamics for departure time choice, and many follow-up studies are possible and warranted.

We formulate a new class of two-person zerosum differential games, in a stochastic setting, where a specification on a target terminal state distribution is imposed on the players. We address such added specification by introducing incentives to the game that guides the players to steer the join distribution accordingly. In the present paper, we only address linear quadratic games with Gaussian target distribution. The solution is characterized by a coupled Riccati equations system, resembling that in the standard linear quadratic differential games. Indeed, once the incentive function is calculated, our problem reduces to a standard one. Tthe framework developed in this paper extends previous results in covariance control, a fast growing research area. On the numerical side, problems herein are reformulated as convex-concave minimax problems for which efficient and reliable algorithms are available.

We formulate a new class of two-person zerosum differential games, in a stochastic setting, where a specification on a target terminal state distribution is imposed on the players. We address such added specification by introducing incentives to the game that guides the players to steer the join distribution accordingly. In the present paper, we only address linear quadratic games with Gaussian target distribution. The solution is characterized by a coupled Riccati equations system, resembling that in the standard linear quadratic differential games. Indeed, once the incentive function is calculated, our problem reduces to a standard one. Tthe framework developed in this paper extends previous results in covariance control, a fast growing research area. On the numerical side, problems herein are reformulated as convex-concave minimax problems for which efficient and reliable algorithms are available.

For the Newtonian 4-body problem in space we prove that any zero angularmomentum bounded solution suffers infinitely many coplanar instants, that is,times at which all 4 bodies lie in the same plane. This result generalizes aknown result for collinear instants ("syzygies") in the zero angular momentumplanar 3-body problem, and extends to the $d+1$ body problem in $d$-space. Theproof, for $d=3$, starts by identifying the center-of-mass zero configurationspace with real $3 \times 3$ matrices, the coplanar configurations withmatrices whose determinant is zero, and the mass metric with the Frobenius(standard Euclidean) norm. Let $S$ denote the signed distance from a matrix tothe hypersurface of matrices with determinant zero. The proof hinges onestablishing a harmonic oscillator type ODE for $S$ along solutions. Bounds oninter-body distances then yield an explicit lower bound $\omega$ for thefrequency of this oscillator, guaranteeing a degeneration within every timeinterval of length $\pi/\omega$. The non-negativity of the curvature oforiented shape space (the quotient of configuration space by the rotationgroup) plays a crucial role in the proof.

Declarative modeling uses symbolic expressions to represent models. With such expressions, one can formalize high-level mathematical computations on models that would be difficult or impossible to perform directly on a lower-level simulation program, in a general-purpose programming language. Examples of such computations on models include model analysis, relatively general-purpose model reduction maps, and the initial phases of model implementation, all of which should preserve or approximate the mathematical semantics of a complex biological model. The potential advantages are particularly relevant in the case of developmental modeling, wherein complex spatial structures exhibit dynamics at molecular, cellular, and organogenic levels to relate genotype to multicellular phenotype. Multiscale modeling can benefit from both the expressive power of declarative modeling languages and the application of model reduction methods to link models across scale. Based on previous work, here we define declarative modeling of complex biological systems by defining the operator algebra semantics of an increasingly powerful series of declarative modeling languages including reaction-like dynamics of parameterized and extended objects; we define semantics-preserving implementation and semantics-approximating model reduction transformations; and we outline a "meta-hierarchy" for organizing declarative models and the mathematical methods that can fruitfully manipulate them.

Synchronization in coupled oscillators networks is a remarkable phenomenon ofrelevance in numerous fields. For Kuramoto oscillators the loss ofsynchronization is determined by a trade-off between coupling strength andoscillator heterogeneity. Despite extensive prior work, the existing sufficientconditions for synchronization are either very conservative or heuristic andapproximate. Using a novel cutset projection operator, we propose a new familyof sufficient synchronization conditions; these conditions rigorously identifythe correct functional form of the trade-off between coupling strength andoscillator heterogeneity. To overcome the need to solve a nonconvexoptimization problem, we then provide two explicit bounding methods, therebyobtaining (i) the best-known sufficient condition for unweighted graphs basedon the 2-norm, and (ii) the first-known generally-applicable sufficientcondition based on the $\infty$-norm. We conclude with a comparative study ofour novel $\infty$-norm condition for specific topologies and IEEE test cases;for most IEEE test cases our new sufficient condition is one to two orders ofmagnitude more accurate than previous rigorous tests.

We experimentally demonstrate that highly structured distributions of workemerge during even the simple task of erasing a single bit. These aresignatures of a refined suite of time-reversal symmetries in distinctfunctional classes of microscopic trajectories. As a consequence, we introducea broad family of conditional fluctuation theorems that the component workdistributions must satisfy. Since they identify entropy production, thecomponent work distributions encode both the frequency of various mechanisms ofsuccess and failure during computing, as well giving improved estimates of thetotal irreversibly-dissipated heat. This new diagnostic tool provides strongevidence that thermodynamic computing at the nanoscale can be constructivelyharnessed. We experimentally verify this functional decomposition and the newclass of fluctuation theorems by measuring transitions between flux states in asuperconducting circuit.

In this paper, tools to study forward invariance properties with robustness to disturbances, referred to as robust forward invariance, are proposed for hybrid dynamical systems modeled as hybrid inclusions. Hybrid inclusions are given in terms of differential and difference inclusions with state and disturbance constraints, for whose definition only four objects are required. The proposed robust forward invariance notions allow for the diverse type of solutions to such systems (with and without disturbances), including solutions that have persistent flows and jumps, that are Zeno, and that stop to exist after finite amount of (hybrid) time. Sufficient conditions for sets to enjoy such properties are presented. These conditions are given in terms of the objects defining the hybrid inclusions and the set to be rendered robust forward invariant. In addition, as special cases, these conditions are exploited to state results on nominal forward invariance for hybrid systems without disturbances. Furthermore, results that provide conditions to render the sublevel sets of Lyapunov-like functions forward invariant are established. Analysis of a controlled inverter system is presented as an application of our results. Academic examples are given throughout this paper to illustrate the main ideas.

In his 1985 paper Sullivan sketched a proof of his structural stabilitytheorem for group actions satisfying certain expansion-hyperbolicity axioms. Inthis paper we relax Sullivan's axioms and introduce a notion of meanderinghyperbolicity for group actions on general metric spaces. This generalizationis substantial enough to encompass actions of certain non-hyperbolic groups,such as actions of uniform lattices in semisimple Lie groups on flag manifolds.At the same time, our notion is sufficiently robust and we prove thatmeandering-hyperbolic actions are still structurally stable. We also prove somebasic results on meandering-hyperbolic actions and give other examples of suchactions.

The main result is Theorem 1: A flow on a connected open set X ⊂ Rd is globally periodic provided (i) periodic points are dense in X, and (ii) at all positive times the flow preserves the partial order defined by a closed convex cone that has nonempty interior and contains no straight line. The proof uses the analog for homeomorphisms due to B. Lemmens et al. [27], a classical theorem of D. Montgomery [31, 32], and a sufficient condition for the nonstationary periodic points in a closed order interval to have rationally related periods (Theorem 2).

Two common types of dynamics, chaotic and monotone, are compared. It is shown that monotone maps in strongly ordered spaces do not have chaotic attractors.

We describe a spectrally filtered discrete-in-time downscaling data assimilation algorithm and prove, in the context of the two-dimensional Navier-Stokes equations, that this algorithm works for a general class of interpolants, such as those based on local spatial averages as well as point measurements of the velocity. Our algorithm is based on the classical technique of inserting new observational data directly into the dynamical model as it is being evolved over time, rather than nudging, and extends previous results in which the observations were defined directly in terms of an orthogonal projection onto the large-scale (lower) Fourier modes. In particular, our analysis does not require the interpolant to be represented by an orthogonal projection, but requires only the interpolant to satisfy a natural approximation of the identity.

Background and objectivesBiological systems with intertwined feedback loops pose a challenge to mathematical modeling efforts. Moreover, rare events, such as mutation and extinction, complicate system dynamics. Stochastic simulation algorithms are useful in generating time-evolution trajectories for these systems because they can adequately capture the influence of random fluctuations and quantify rare events. We present a simple and flexible package, BioSimulator.jl, for implementing the Gillespie algorithm, τ-leaping, and related stochastic simulation algorithms. The objective of this work is to provide scientists across domains with fast, user-friendly simulation tools.MethodsWe used the high-performance programming language Julia because of its emphasis on scientific computing. Our software package implements a suite of stochastic simulation algorithms based on Markov chain theory. We provide the ability to (a) diagram Petri Nets describing interactions, (b) plot average trajectories and attached standard deviations of each participating species over time, and (c) generate frequency distributions of each species at a specified time.ResultsBioSimulator.jl's interface allows users to build models programmatically within Julia. A model is then passed to the simulate routine to generate simulation data. The built-in tools allow one to visualize results and compute summary statistics. Our examples highlight the broad applicability of our software to systems of varying complexity from ecology, systems biology, chemistry, and genetics.ConclusionThe user-friendly nature of BioSimulator.jl encourages the use of stochastic simulation, minimizes tedious programming efforts, and reduces errors during model specification.

Coherent structures form spontaneously in nonlinear spatiotemporal systems and are found at all spatial scales in natural phenomena from laboratory hydrodynamic flows and chemical reactions to ocean, atmosphere, and planetary climate dynamics. Phenomenologically, they appear as key components that organize the macroscopic behaviors in such systems. Despite a century of effort, they have eluded rigorous analysis and empirical prediction, with progress being made only recently. As a step in this, we present a formal theory of coherent structures in fully discrete dynamical field theories. It builds on the notion of structure introduced by computational mechanics, generalizing it to a local spatiotemporal setting. The analysis' main tool employs the local causal states, which are used to uncover a system's hidden spatiotemporal symmetries and which identify coherent structures as spatially localized deviations from those symmetries. The approach is behavior-driven in the sense that it does not rely on directly analyzing spatiotemporal equations of motion, rather it considers only the spatiotemporal fields a system generates. As such, it offers an unsupervised approach to discover and describe coherent structures. We illustrate the approach by analyzing coherent structures generated by elementary cellular automata, comparing the results with an earlier, dynamic-invariant-set approach that decomposes fields into domains, particles, and particle interactions.

Let X be a connected open set in n-dimensional Euclidean space, partiallyordered by a closed convex cone K with nonempty interior: y > x if and only ify-x is nonzero and in K. Theorem: If F is a monotone local flow in X whose periodic points are densein X, then F is globally periodic.