Discrete Transitions Research Articles

Open Markov Type Population Models: From Discrete to Continuous Time

We address the problem of finding a natural continuous time Markov type process—in open populations—that best captures the information provided by an open Markov chain in discrete time which is usually the sole possible observation from data. Given the open discrete time Markov chain, we single out two main approaches: In the first one, we consider a calibration procedure of a continuous time Markov process using a transition matrix of a discrete time Markov chain and we show that, when the discrete time transition matrix is embeddable in a continuous time one, the calibration problem has optimal solutions. In the second approach, we consider semi-Markov processes—and open Markov schemes—and we propose a direct extension from the discrete time theory to the continuous time one by using a known structure representation result for semi-Markov processes that decomposes the process as a sum of terms given by the products of the random variables of a discrete time Markov chain by time functions built from an adequate increasing sequence of stopping times.

Generating an Adaptive and Robust Walking Pattern for a Prosthetic Ankle-Foot by Utilizing a Nonlinear Autoregressive Network With Exogenous Inputs.

One of the major challenges in developing powered lower limb prostheses is emulating the behavior of an intact lower limb with different walking speeds over diverse terrains. Numerous studies have been conducted on control algorithms in the field of rehabilitation robotics to achieve this overarching goal. Recent studies on powered prostheses have frequently used a hierarchical control scheme consisting of three control levels. Most control structures have at least one element of discrete transition properties that requires numerous sensors to improve classification accuracy, consequently increasing computational load and costs. In this study, we proposed a user-independent and free-mode method for eliminating the need to switch among different controllers. We constructed a database by using four OPAL wearable devices (Mobility Lab, APDM Inc., USA) for seven able-bodied subjects. We recorded the gait of each subject at three ambulation speeds during ground-level walking to train a nonlinear autoregressive network with an exogenous input recurrent neural network (NARX RNN) to estimate foot orientation (angular position) in the sagittal plane using shank angular velocity as external input. The trained NARX RNN estimated the foot orientation of all the subjects at different walking speeds over flat terrain with an average root-mean-square error (RMSE) of 2.1° ± 1.7°. The minimum correlation between the estimated and measured values was 86%. Moreover, a t-test showed that the error was normally distributed with a high certainty level (0.88 minimum p -value).

Heider and coevolutionary balance: From discrete to continuous phase transition.

Structural balance in social complex networks has been modeled with two types of triplet interactions. First is the interaction that only considers the dynamic role for links or relationships (Heider balance), and second is the interaction that considers both individual opinions (nodes) and relationships in network dynamics (coevolutionary balance). The question is, as the temperature varies, which is a measure of the average irrationality of individuals in a society, how structural balance can be created or destroyed by each of these triplet interactions. We use statistical mechanics methods and observe through analytical calculation and numerical simulation that unlike the Heider balance triplet interaction which has a discrete phase transition, the coevolutionary balance has a continuous phase transition. The critical temperature of the presented model changes with the root square of the network size, which is a linear dependence in the thermal Heider balance.

Mean-field solution for critical behavior of signed networks in competitive balance theory.

The competitive balance model has been proposed as an extension to the balance model to address the conflict of interests in signed networks. In this model, two different paradigms or interests compete with each other to dominate the network's relations and impose their own values. In this paper, using the mean-field method, we examine the thermal behavior of the competitive balance model. Our results show that under a certain temperature, the symmetry between two competing interests will spontaneously break which leads to a discrete phase transition. So, starting with a heterogeneous signed network, if agents aim to decrease tension stemming from competitive balance theory, evolution ultimately chooses only one of the existing interests and stability arises where one paradigm dominates the system. The critical temperature depends linearly on the number of nodes, which is a linear dependence in the thermal balance theory as well. Finally, the results obtained through the mean-field method are verified by a series of simulations.

Preliminary Flow Modeling by Hybrid Automata Alternating Continuous Reaction and Discrete Transit for Pharmacokinetics.

In traditional pharmacokinetic models, blood flow or liquid transit is often expressed as first-order kinetics. When the flow expression by first-order kinetics is used for dynamic simulation, the flow velocity illogically depends on the step size of a solver of ordinary differential equations. In this study, we propose flow modeling using hybrid automata that combine ordinary differential equations and recursive equations, and we have preliminarily applied the constructed models to several examples. The blood concentration-time profiles of p-aminohippurate and propranolol after intravenous administration were successfully reproduced by simple hybrid automata. The simulation results of one-dimensional tube flow have demonstrated that the fluid velocity in the hybrid automata was independent of the step size of the ordinary differential equation solver. A body fluid model coordinated various flows in a human body with scheduled daily activities and could be used as a drug container to describe formulation-dependent disposition of 5-aminosalicylic acid and enterohepatic circulation of a virtual drug. These findings suggested that flow modeling using hybrid automata could avoid the logical inconsistency in the traditional pharmacokinetic modeling and that the hybrid automata have high versatility and a wide range of applicability to pharmacokinetic analysis. Because our method can define various intervals for multiple recursive equations, the resolution of a specific part of a model can be adjusted relatively freely while the whole body is being roughly modeled, which would be beneficial to refine a coarse model into a fine model in future. SIGNIFICANCE STATEMENT: There is a logical inconsistency in flow expression by first-order kinetics in ordinary differential equations used in traditional pharmacokinetic modeling. It is difficult to model a whole human body using flow models in partial differential equations because of the excessive calculation costs. Our simulations on tube flow and body fluids have demonstrated that the flow modeling using hybrid automata could avoid the problems. The preliminary applications of hybrid automata to several examples highlighted its high versatility in pharmacokinetic analysis.

Division of paraffin melting zone based on multiscale experiments

Phase change energy storage materials are widely used in the field of renewable energy. Paraffin is one of the common phase change energy storage materials. As a multi-component hydrocarbon mixture, the melting of paraffin is different from that of pure substance. In addition the solid and liquid zones, there is also a fuzzy zone in which solid and liquid coexist. In this paper, the melting characteristics of paraffin in phase transition zone are studied by multi-scale experiments. Through the visualization experiment of square cavity paraffin melting, the solid zone, fuzzy zone and liquid zone are determined, and the moving process of phase interface is tracked by digital pictures and infrared heat maps. The evolution process of the pore structure in the fuzzy zone under different temperatures is photographed by means of the micro-experiment, and it is revealed that there are two areas in the fuzzy zone, porous media area and multi-phase flow area. The results show that the melting process of paraffin can be divided into four zones: liquid zone, multi-phase flow zone, porous media zone, and solid phase zone. According to the polarizing optical microscopy picture, the continuous phase and discrete phase transition relationship between solid wax crystal and liquid paraffin is captured. The polarizing optical microscopy picture is statistically analyzed, and the critical liquid phase ratio of the transition from porous media area to multi-phase flow area is given under experimental conditions.

Measuring inconsistency in some branching time logics

Branching time logics have been studied in computer science since the 1980s primarily to model the tree of computations for discrete transition systems. Inconsistency measures for propositional logic have been studied since the early 2000s by AI researchers. This paper introduces inconsistency measures for three branching time logics: ABTL, BBTL, and CBTL. In order to measure inconsistency properly in branching time logics, the semantics differs from the standard semantics using the construction of a canonical tree. ABTL extends propositional logic by three pairs of unary operators: next time, path, and eventually. BBTL adds a pair of binary Until operators that can be applied only to propositional logic formulas. CBTL adds the propositional connectives to BBTL formulas. Propositional logic inconsistency measures are extended to these logics and examples of the computation are given. These measures are shown to satisfy intuitively the concepts involved in the branching time logic operators unlike direct extensions of propositional logic inconsistency measures.

Stability analysis of a rotor system with electromechanically coupled boundary conditions under periodic axial load

The parametric instability of a rotor system with electromechanically coupled boundary conditions under periodic axial loads is studied. Based on the current flowing piezoelectric shunt damping technique, the detailed rotor model is established by the finite element (FE) method. In the matrix assembly procedure, a novel simple process is proposed to make the equations of shunt circuits more conveniently to be introduced into the global FE equations. The discrete state transition matrix method which is used for determining the influence of circuit parameters on instability regions in this paper has also been presented. The numerical simulation shows that only the combination instability regions exist when the shaft is rotating. The mechanical damping has different effect on the simple and combined instability regions. These two points are consistent with the previous references, which verifies the obtained FE model. In addition, the simulated results also reveal that the introduction of shunt circuits has little influence on the rotor’s original whirling frequencies. It gives rise to the appearance of new synchronous whirl modes. The new whirling frequencies are combined with the original ones to form the new combination instability regions. Furthermore, the resistance of shunt circuits has the same performance as the mechanical damping has. That is, moving up the start points of instability regions and expanding its width.

A Continuous Time Representation of smFRET for the Extraction of Rapid Kinetics

Single molecule Förster resonance energy transfer (smFRET) experiments are a powerful way to probe switching kinetics between the conformational states of a molecule where a portion of the molecule is attached to a donor fluorophore and another to an acceptor fluorophore. In order to learn the kinetic rates of a molecule from smFRET data, hidden Markov models (HMMs) are often used. HMMs treat time traces in discrete time even though molecule kinetics evolve continuously in time. In addition, HMMs inherently assume that the probed kinetic rates are slower than the data acquisition rate. Therefore, if the smFRET data acquisition rate is not sufficiently high, then HMMs become inappropriate for extracting the molecule's switching kinetics. Here, we would like to learn kinetic rates from smFRET data even if switching kinetics are faster than the data acquisition rate. To achieve our goal of inferring the kinetic rates, we use a variant of a hidden Markov jump process (HMJP), for the analysis of parallel measurements from donor and acceptor channels. Our HMJP approach generalizes HMM in two critical ways: (1) it models the molecule as switching between different conformational states in continuous time and (2) it allows us to learn the kinetic rates without approximating them as discrete transition probabilities nor assuming that the rates are slower than the data acquisition rate. In our HMJP setup, we infer switching kinetics and photon emission rates that go beyond the dynamics regimes that HMMs can deal with. The robustness of our method is validated on simulated data and we demonstrated its performance on experimental data from FRET labeled Holliday junctions.

Mechanistic insights and selected synthetic routes of atomically precise metal nanoclusters

AbstractDuring the last few decades, noble metal nanoclusters (NCs) have become an exciting building block in the field of nanoscience. With their ultrasmall size that ranges between 1 and 2 nm, NCs fill the gap between atoms and nanoparticles (NPs), and they show significantly different physicochemical properties compared to their bulk counterparts, such as molecule‐like HOMO‐LUMO discrete electronic transitions, photoluminescence, etc. These properties made NCs potential candidates in various applications, including catalysis, chemical/bioimaging, biomedicine, sensing, and energy conversion. Controlling the size of NPs, which usually exhibit a degree of polydispersity, has been a significant challenge for nano‐scientists. However, metal NCs with atomic precision pave the way to accurately fabricate NPs based on an atom‐by‐atom assembly. This Perspective is directed to the community of nano‐scientists interested in the field of NCs and summarizes the most commonly used synthetic routes of atomically precise metal NCs. Moreover, this Perspective provides an understanding of the different techniques used to control the size of metal NCs with insights on switching the surface ligands from phosphine to thiol. This Perspective also explains the role of physicochemical parameters in different synthetic routes such as high‐temperature route, CO‐directed route, solid‐state route, ligand‐exchange‐induced size/structure transformation (LEIST), etc. We finally give a brief outlook on future challenges of currently used synthetic routes with some suggestions to improve them.

EM algorithm for stochastic hybrid systems

A stochastic hybrid system, also known as a switching diffusion, is a continuous-time Markov process with state space consisting of discrete and continuous parts. We consider parametric estimation of the Q matrix for the discrete state transitions and of the drift coefficient for the diffusion part. First, we derive the likelihood function under the complete observation of a sample path in continuous-time. Then, extending a finite-dimensional filter for hidden Markov models developed by Elliott et al. (Hidden Markov Models, Springer, 1995) to stochastic hybrid systems, we derive the likelihood function and the EM algorithm under a partial observation where the continuous state is monitored continuously in time, while the discrete state is unobserved.

A homoleptic alkynyl-protected [Ag9Cu6( t BuC[triple bond, length as m-dash]C)12]+ superatom with free electrons: synthesis, structure analysis, and different properties compared with the Au7Ag8 cluster in the M15 + series.

We report the first homoleptic alkynyl-protected AgCu superatomic nanocluster [Ag9Cu6(tBuCC)12]+ (NC 1, also Ag9Cu6 in short), which has a body-centered-cubic structure with a Ag1@Ag8@Cu6 metal core. Such a configuration is reminiscent of the reported AuAg bimetallic nanocluster [Au1@Ag8@Au6(tBuCC)12]+ (NC 2, also Au7Ag8 in short), which is also synthesized by an anti-galvanic reaction (AGR) approach with a very high yield for the first time in this study. Despite a similar Ag8 cube for both NCs, structural anatomy reveals that there are some subtle differences between NCs 1 and 2. Such differences, plus the different M1 kernel and M6 octahedron, lead to significantly different optical absorbance features for NCs 1 and 2. Density functional theory calculations revealed the LUMO and HOMO energy levels of NCs 1 and 2, where the characteristic absorbance peaks can be correlated with the discrete molecular orbital transitions. Finally, the stability of NCs 1 and 2 at different temperatures, in the presence of an oxidant or Lewis base, was investigated. This study not only enriches the M15+ series, but also sets an example for correlating the structure–property relationship in alkynyl-protected bimetallic superatomic clusters.

Gauge invariant and gauge dependent aspects of topological walking colloidal bipeds.

Paramagnetic colloidal spheres assemble to colloidal bipeds of various length in an external magnetic field. When the bipeds reside above a magnetic pattern and we modulate the direction of the external magnetic field, the rods perform topologically distinct classes of protected motion above the pattern. The topological protection allows each class to be robust against small continuous deformations of the driving loop of the external field. We observe motion of the rod from a passive central sliding and rolling motion for short bipeds toward a walking motion with both ends of the rod alternately touching down on the pattern for long bipeds. The change of character of the motion occurs in form of discrete topological transitions. The topological protection makes walking a form of motion robust against the breaking of the non symmorphic symmetry. In patterns with non symmorphic symmetry walking is reversible. In symmorphic patterns lacking a glide plane the walking can be irreversible or reversible involving or not involving ratchet jumps. Using different gauges allows us to unravel the active and passive aspects of the topological walks.

Smooth Approximations for Hybrid Optimal Control Problems with Application to Robotic Walking

Abstract This paper investigates optimal control problems formulated over a class of hybrid dynamical systems which display event-triggered discrete jumps. Due to the discontinuous nature of the underlying dynamics, previous approaches to solving optimal control problems over this class of systems generally rely on fixing the number and sequence of discrete jumps a priori, or search over possible mode sequences in a combinatorial manner. Employing contributions from the geometric theory of hybrid systems, we instead formulate a family of smooth approximate problems formulated over a family of smooth control systems which faithfully approximate the dynamics of the original hybrid system, in an appropriate metric. Efficient gradient-based methods can be used to solve the smooth approximations, without specifying the sequence of discrete transitions ahead of time. Under appropriate hypothesis, the gradients of the smooth problem are shown to be well-conditioned and closely approximate the gradients of the non-smooth problem (when they exist). Two cases studies demonstrate the utility of the approach, including an in-depth application to generating a stable walking motion for a bipedal robot.

Quantum electromechanics with levitated nanoparticles

Preparing and observing quantum states of nanoscale particles is a challenging task with great relevance for quantum technologies and tests of fundamental physics. In contrast to atomic systems with discrete transitions, nanoparticles exhibit a practically continuous absorption spectrum and thus their quantum dynamics cannot be easily manipulated. Here, we demonstrate that charged nanoscale dielectrics can be artificially endowed with a discrete level structure by coherently interfacing their rotational and translational motion with a superconducting qubit. We propose a pulsed scheme for the generation and read-out of motional quantum superpositions and entanglement between several levitated nanoparticles, providing an all-electric platform for networked hybrid quantum devices.

Analysis of resolution of tomographic-type reconstruction from discrete data for a class of distributions * *This work was supported in part by NSF Grants DMS-1615124 and DMS-1906361.

Let f(x), , be a piecewise smooth function with a jump discontinuity across a smooth surface . Let f Λϵ denote the Lambda tomography (LT) reconstruction of f from its discrete Radon data . The sampling rate along each variable is ∼ϵ. First, we compute the limit for a generic . Once the limiting function is known (which we call the discrete transition behavior, or DTB for short), the resolution of reconstruction can be easily found. Next, we show that straight segments of lead to non-local artifacts in f Λϵ , and that these artifacts are of the same strength as the useful singularities of f Λϵ . We also show that f Λϵ (x) does not converge to its continuous analogue f Λ = (−Δ)1/2 f as ϵ → 0 even if . Results of numerical experiments presented in the paper confirm these conclusions. We also consider a class of Fourier integral operators with the same canonical relation as the classical Radon transform adjoint, and a class of distributions , , and obtain easy to use formulas for the DTB when is computed from discrete data . Exact and LT reconstructions are particular cases of this more general theory.

N-Alkanes to n-alcohols: Formal primary C─H bond hydroxymethylation via quadruple relay catalysis.

Nature is able to synergistically combine multiple enzymes to conduct well-ordered biosynthetic transformations. Mimicking nature's multicatalysis in vitro may give rise to new chemical transformations via interplay of numerous molecular catalysts in one pot. The direct and selective conversion of abundant n-alkanes to valuable n-alcohols is a reaction with enormous potential applicability but has remained an unreached goal. Here, we show that a quadruple relay catalysis system involving three discrete transition metal catalysts enables selective synthesis of n-alcohols via n-alkane primary C─H bond hydroxymethylation. This one-pot multicatalysis system is composed of Ir-catalyzed alkane dehydrogenation, Rh-catalyzed olefin isomerization and hydroformylation, and Ru-catalyzed aldehyde hydrogenation. This system is further applied to synthesis of α,ω-diols from simple α-olefins through terminal-selective hydroxymethylation of silyl alkanes.

General methods and properties to evaluate continuum limits of the 1D discrete time quantum walk

Models of quantum walks which admit continuous time and continuous spacetime limits have recently led to quantum simulation schemes for simulating fermions in relativistic and nonrelativistic regimes (Molfetta GD, Arrighi P. A quantum walk with both a continuous-time and a continuous-spacetime limit, 2019). This work continues the study of relationships between discrete time quantum walks (DTQW) and their ostensive continuum counterparts by developing a more general framework than was done in Molfetta and Arrighi (A quantum walk with both a continuous-time and a continuous-spacetime limit, 2019) to evaluate the continuous time limit of these discrete quantum systems. Under this framework, we prove two constructive theorems concerning which internal discrete transitions (“coins”) admit nontrivial continuum limits. We additionally prove that the continuous space limit of the continuous time limit of the DTQW can only yield massless states which obey the Dirac equation. Finally, we demonstrate that for general coins the continuous time limit of the DTQW can be identified with the canonical continuous time quantum walk when the coin is allowed to transition through the continuous limit process.

On the Performance of Kalman Filter for Markov Jump Linear Systems with Mode Mismatch

Markov jump linear system (MJLS) is a class of hybrid systems where the continuous states evolve linearly and the discrete state transitions are modeled via a Markov chain. It has been shown that the optimal estimator for MJLS is a mode-based Kalman filter when the discrete mode is given. However, in practical applications, there are situations where the mode information is inaccurate and mode mismatches result in a biased estimate from the mode-based Kalman filter. This paper, for the first time, studies the impact of time correlated mode mismatch errors on MJLS state estimation using a mode-based Kalman filter. Unlike prior efforts that (1) assume mode mismatches are independent and identically distributed and (2) are limited to bi-modal systems, this paper is built on a general MJLS setup with a Markovian model for mode mismatch errors. The main contribution of this work lies in deriving the statistics of the bias term from a mode-based Kalman filter estimation with the aforementioned system setup. The sufficient conditions based on the probabilities of mode mismatch errors for the bias to be statistically convergent are derived. Since time correlated mode mismatch errors can effectively capture communication link impairments in a cyber-physical system, this new fundamental result provides guidance on the design of estimation strategies for MJLS and sheds light on their resilience in the presence of mode mismatch errors.

Synthesis of sensor deception attacks at the supervisory layer of Cyber–Physical Systems

We study the security of Cyber–Physical Systems (CPS) in the context of the supervisory control layer. Specifically, we propose a general model of a CPS attacker in the framework of discrete event systems and investigate the problem of synthesizing an attack strategy for a given feedback control system. Our model captures a class of deception attacks, where the attacker has the ability to hijack a subset of sensor readings and mislead the supervisor, with the goal of inducing the system into an undesirable state. We utilize a game-like discrete transition structure, called Insertion–Deletion Attack structure (IDA), to capture the interaction between the supervisor and the environment (which includes the system and the attacker). We show how to use IDAs to synthesize three different types of successful stealthy attacks, i.e., attacks that avoid detection from the supervisor and cause damage to the system.