Discrete Time Research Articles

Growing integer partitions with uniform marginals and the equivalence of partition ensembles

We present an explicit construction of a Markovian random growth process on integer partitions such that given it visits some level n, it passes through any partition λ of n with equal probabilities. The construction has continuous time, but we also investigate its discrete time jump chain. The jump probabilities are given by explicit but complicated expressions, so we find their asymptotic behavior as the partition becomes large. This allows us to explain how the limit shape is formed.Using the known connection of the considered probabilistic objects to Poisson point processes, we give an alternative description of the partition growth process in these terms. Then we apply the constructed growth process to find sufficient conditions for a phenomenon known as equivalence of two ensembles of random partitions for a finite number of partition characteristics. This result allows to show that counts of odd and even parts in a random partition of n are asymptotically independent as n→∞ and to find their limiting distributions, which are, somewhat surprisingly, different.

Artificial intelligence-enhanced electrocardiography predicts 10-year risk of atherosclerotic cardiovascular disease

Abstract Background The ACC/AHA pooled cohort equations (PCE) calculates the 10-year primary risk of atherosclerotic cardiovascular disease (ASCVD), an indication to initiate primary prevention statin therapy in international guidelines. The validity of PCE is limited by its overestimation of risk in high-risk cohorts and an ethnic heterogeneity. Electrocardiography (ECG) has not been incorporated in current ASCVD risk estimation tools. Artificial Intelligence (AI)-based models are capable to capture patterns that are informative for cardiovascular prognosis. Purpose We explored the predictive value of a discrete-time survival model in 10-year ASCVD risk in comparison to PCE. Methods We developed an AI-ECG model to predict the 10-year ASCVD risk in an unselected secondary care population from the USA comprised of 1,163,401 ECGs from 189,539 patients. Data was split into training/validation and hold-out test sets by patient ID. Our AI-ECG model uses deep learning with a residual convolutional neural network with a discrete-time survival loss function to output subject-specific survival curves, accounting for both time to event and censorship (i.e., loss to follow up). We compared the performance of out AI-ECG model using a subset of outpatient data (n=4590) of the BIDMC dataset to PCE and ASCVD risk factors including systolic blood pressure, total cholesterol, HDL cholesterol, hypertension, smoking history, diabetes mellitus, and ethnicity. Performances were evaluated using cox survival analysis. Our AI-ECG model was externally validated in the UKB Biobank (UKB), where we compared the performance of our AI-ECG model to predictions of existing AI-ECG models such as the Stanford Estimator of Electrocardiogram Risk (SEER) model. Results Our AI-ECG successfully predicted future ASCVD events in individuals (C-index 0.721 (0.719-0.723), 5-year AUC 0.761 (0.758-0.763), and differentiated between high and low risk group with an adjusted HR of 2.33 (2.26-2.41). Upon internal validation our AI-ECG model showed a superior performance (C-index 0.679 (0.651-0.708)) than PCE (0.605 (0.577-0.634)) and all risk factors combined (C index 0.642 (0.613-0.672)). When combining AI-ECG predictions with all risk factors, the model showed additional predictive value (0.686 (0.657-0.715)) (Figure 1). Our AI-ECG model showed modest predictive value within the UKB dataset (C-index 0.655 (0.637-0.673)), but significantly outperformed the SEER model (C-index 0.547 (0.527-0.567), p <0.0001 for comparison with AIRE). Conclusion We have shown the potential of AI-ECG models in predicting future ASCVD risks. AI-ECG models demonstrated accurate discrete time survival prediction and good classification performance superior to that of the SEER model, as well as PCE and ASCVD risk factors. . Accurate risk assessment of ASCVD may help prevent adverse events in high risk subjects and save unnecessary pharmacological interventios in low risk subjects.Figure 1

Embedding and elimination for performance analysis in stochastic process algebra dtsdPBC

ABSTRACT dtsdPBC extends the well-known algebra of parallel processes, Petri box calculus (PBC), by incorporating discrete time stochastic and deterministic delays. To analyze performance in this extended calculus, the underlying semi-Markov chains, and the related (complete) and reduced discrete time Markov chains of the process expressions are built. The semi-Markov chains are extracted using the embedding method, which constructs the embedded discrete time Markov chains and calculates the sojourn time distributions in the states. The reductions of the discrete time Markov chains are obtained through the elimination method, which removes the vanishing states (those with zero sojourn times) and recalculates the transition probabilities among the tangible states (those with positive sojourn times). We prove that the reduced semi-Markov chain coincides with the reduced discrete time Markov chain, by demonstrating that an additional embedding into the reduced semi-Markov chain is needed for the reduced embedded discrete time Markov chain to match the embedded reduced discrete time Markov chain, and by comparing the respective sojourn times.

Noise-Induced Quantum Synchronization and Entanglement in a Quantum Analogue of Huygens' Clock.

We propose a quantum analogue of the Huygens clock, where the phases of two spins synchronize through their interaction with a shared environment. This environment acts like the escapement mechanism in a mechanical clock, regulating the gear train and allowing discrete timing advances. In our model, the relative phases of the two spins synchronize via a mutually correlated environment. We demonstrate that several arguments can significantly reduce the cardinality of the allowed measurements for a system of qubits, thus simplifying the problem. We present a numerically efficient method to calculate the degree of quantumness in the correlations of the final density matrix, providing a tight upper bound for rank 3 and rank 4 density matrices. We suggest a potential realization of noise-induced synchronization between two nuclear spins coupled to a common ancilla undergoing dynamical decoupling.

Second order conservative Lagrangian DG schemes for compressible flow and their application in preserving spherical symmetry in two-dimensional cylindrical geometry

In this paper, we construct a class of second-order cell-centered Lagrangian discontinuous Galerkin (DG) schemes for the two-dimensional compressible Euler equations on quadrilateral meshes. This Lagrangian DG scheme is based on the physical coordinates rather than the fixed reference coordinates, hence it does not require studying the evolution of the Jacobian matrix for the flow mapping between the different coordinates. The conserved variables are solved directly, and the scheme can preserve the conservation property for mass, momentum and total energy. The strong stability preserving (SSP) Runge-Kutta (RK) method is used for the time discretization. Furthermore, there are two main contributions. Firstly, differently from the previous work, we design a new Lagrangian DG scheme which is truly second-order accurate for all the variables such as density, momentum, total energy, pressure and velocity, while the similar DG schemes in the literature may lose second-order accuracy for certain variables, as shown in numerical experiments. Secondly, as an extension and application, we develop a particular Lagrangian DG scheme in the cylindrical geometry, which is designed to be able to preserve one-dimensional spherical symmetry for all the linear polynomials in two-dimensional cylindrical coordinates when computed on an equal-angle-zoned initial grid. The distinguished feature is that it can maintain both the spherical symmetry and conservation properties, which is very important for many applications such as implosion problems. A series of numerical experiments in the two-dimensional Cartesian and cylindrical coordinates are given to demonstrate the good performance of the Lagrangian DG schemes in terms of accuracy, symmetry and non-oscillation.

On a drift‐diffusion model for perovskite solar cells

AbstractWe introduce a vacancy‐assisted charge transport model for perovskite solar cells. This instationary drift‐diffusion system describes the motion of electrons, holes, and ionic vacancies and takes into account Fermi–Dirac statistics for electrons and holes and the Fermi–Dirac integral of order for the mobile ionic vacancies in the perovskite. The free energy functional we work with corresponds to that choice of the statistical relations. To verify the existence of weak solutions, we consider a problem with regularized state equations and reaction terms on any arbitrarily chosen finite time interval. We motivate its solvability by time discretization and passage to the time‐continuous limit. A priori estimates for the chemical potentials that are independent of the regularization level ensure the existence of solutions to the original problem. These types of estimates rely on Moser iteration techniques and can also be obtained for solutions to the original problem.

Relaxed Equilibria for Time-Inconsistent Markov Decision Processes

This paper considers an infinite-horizon Markov decision process (MDP) that allows for general nonexponential discount functions in both discrete and continuous time. Because of the inherent time inconsistency, we look for a randomized equilibrium policy (i.e., relaxed equilibrium) in an intrapersonal game between an agent’s current and future selves. When we modify the MDP by entropy regularization, a relaxed equilibrium is shown to exist by a nontrivial entropy estimate. As the degree of regularization diminishes, the entropy-regularized MDPs approximate the original MDP, which gives the general existence of a relaxed equilibrium in the limit by weak convergence arguments. As opposed to prior studies that consider only deterministic policies, our existence of an equilibrium does not require any convexity (or concavity) of the controlled transition probabilities and reward function. Interestingly, this benefit of considering randomized policies is unique to the time-inconsistent case.

Möbius Invariant Y-systems (Cluster Structures) for Miquel Dynamics

Abstract Miquel dynamics is a discrete time dynamics for circle patterns, which relies on Miquel’s six circle theorem. Previous work shows that the evolution of the circle centers satisfy the dSKP equation on the octahedral lattice $A_{3}$. As a consequence, Miquel dynamics is a discrete integrable system. Moreover, Miquel dynamics give rise to a real-valued cluster structure. The evolution of the cluster variables under Miquel dynamics is also called a Y-system in the discrete integrable systems community. If the Y-system is real positive-valued then the circle pattern is accompanied by an invariant dimer model, an exactly solvable model studied in statistical physics. However, while circle patterns are Möbius invariant, the circle centers and the Y-system are not Möbius invariant, which violates the so called transformation group principle. In this article we show that half the intersection points satisfy the dSKP equation as well, and we introduce two new real-valued Y-systems for Miquel dynamics that involve only the intersection points. Therefore, the new Y-systems are Möbius invariant, and thus satisfy the transformation group principle. We also show that the circle centers and intersection points combined satisfy the dSKP equation on the 4-dimensional octahedral lattice $A_{4}$. In addition, we present two more complex-valued Y-systems for Miquel dynamics, which are real-valued in and only in the case of integrable circle patterns. We also investigate the special cases of harmonic embeddings and s-embeddings, which relate to the spanning tree and Ising model, respectively.

Closed-Boundary Reflections of Shallow Water Waves as an Open Challenge for Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) have recently emerged as a promising alternative to traditional numerical methods for solving partial differential equations (PDEs) in fluid dynamics. By using PDE-derived loss functions and auto-differentiation, PINNs can recover solutions without requiring costly simulation data, spatial gridding, or time discretization. However, PINNs often exhibit slow or incomplete convergence, depending on the architecture, optimization algorithms, and complexity of the PDEs. To address these difficulties, a variety of novel and repurposed techniques have been introduced to improve convergence. Despite these efforts, their effectiveness is difficult to assess due to the wide range of problems and network architectures. As a novel test case for PINNs, we propose one-dimensional shallow water equations with closed boundaries, where the solutions exhibit repeated boundary wave reflections. After carefully constructing a reference solution, we evaluate the performance of PINNs across different architectures, optimizers, and special training techniques. Despite the simplicity of the problem for classical methods, PINNs only achieve accurate results after prohibitively long training times. While some techniques provide modest improvements in stability and accuracy, this problem remains an open challenge for PINNs, suggesting that it could serve as a valuable testbed for future research on PINN training techniques and optimization strategies.

An R Package for Nonparametric Inference on Dynamic Populations with Infinitely Many Types.

Fleming-Viot diffusions are widely used stochastic models for population dynamics that extend the celebrated Wright-Fisher diffusions. They describe the temporal evolution of the relative frequencies of the allelic types in an ideally infinite panmictic population, whose individuals undergo random genetic drift and at birth can mutate to a new allelic type drawn from a possibly infinite potential pool, independently of their parent. Recently, Bayesian nonparametric inference has been considered for this model when a finite sample of individuals is drawn from the population at several discrete time points. Previous works have fully described the relevant estimators for this problem, but current software is available only for the Wright-Fisher finite-dimensional case. Here, we provide software for the general case, overcoming some nontrivial computational challenges posed by this setting. The R package FVDDPpkg efficiently approximates the filtering and smoothing distribution for Fleming-Viot diffusions, given finite samples of individuals collected at different times. A suitable Monte Carlo approximation is also introduced in order to reduce the computational cost.

Topological transport of a classical droplet in a lattice of time

AbstractThouless charge pumps are quantum mechanical devices whose operation relies on topology. They provide the means for transporting quantum matter in space lattices with a single quantum precision. Contrasting space crystals that spontaneously break a continuous spatial translation symmetry and form crystals in space, time crystals have emerged as novel states of matter that organise into time lattices and spontaneously break a discrete time translation symmetry. The utility of Thouless pumps that enable topologically protected quantised transport of electrons and neutral atoms in spatial superlattices leads to the question if corresponding devices exist for time crystals. Here we show that topological pumps can be realised for time solids by transporting droplets of a liquid forward and backward in time lattices and we measure the topological index that characterises such pumping processes. By exploiting a synthetic time dimension, classical time crystals can circumvent the quantum tunnelling that underpins Thouless charge pumps. Our results establish topological pumping through time instead of space and pave the way for applications of time crystals.Key Points Periodically driven fluid droplets can be made to bounce persistently in a gravitational field and may undergo spontaneous time translation symmetry breaking forming classical time crystals. These classical time crystals can be pumped from one time site to another, and this pumping process is characterised by a topological index. Our observations establish a classical temporal counterpart to Thouless pumping of quantum particles in space lattices.

Dynamics of finite width Kernel and prediction fluctuations in mean field neural networks*

Abstract We analyze the dynamics of finite width effects in wide but finite feature learning neural networks. Starting from a dynamical mean field theory description of infinite width deep neural network kernel and prediction dynamics, we provide a characterization of the O ( 1 / width ) fluctuations of the dynamical mean field theory order parameters over random initializations of the network weights. Our results, while perturbative in width, unlike prior analyses, are non-perturbative in the strength of feature learning. We find that once the mean field/µP parameterization is adopted, the leading finite size effect on the dynamics is to introduce initialization variance in the predictions and feature kernels of the networks. In the lazy limit of network training, all kernels are random but static in time and the prediction variance has a universal form. However, in the rich, feature learning regime, the fluctuations of the kernels and predictions are dynamically coupled with a variance that can be computed self-consistently. In two layer networks, we show how feature learning can dynamically reduce the variance of the final tangent kernel and final network predictions. We also show how initialization variance can slow down online learning in wide but finite networks. In deeper networks, kernel variance can dramatically accumulate through subsequent layers at large feature learning strengths, but feature learning continues to improve the signal-to-noise ratio of the feature kernels. In discrete time, we demonstrate that large learning rate phenomena such as edge of stability effects can be well captured by infinite width dynamics and that initialization variance can decrease dynamically. For convolutional neural networks trained on CIFAR-10, we empirically find significant corrections to both the bias and variance of network dynamics due to finite width.

Insider trading in discrete time Kyle games

AbstractWe present a new discrete time version of Kyle’s (Econometrica 53(6):1315–1335, 1985) classic model of insider trading, formulated as a generalised extensive form game. The model has three kinds of traders: an insider, random noise traders, and a market maker. The insider aims to exploit her informational advantage and maximise expected profits while the market maker observes the total order flow and sets prices accordingly. First, we show how the multi-period model with finitely many pure strategies can be reduced to a (static) social system in the sense of Debreu (Proc Natl Acad Sci 38(10):886–893, 1952) and prove the existence of a sequential Kyle equilibrium, following Kreps and Wilson (Econometrica 50(4):863–894, 1982). This works for any probability distribution with finite support of the noise trader’s demand and the true value, and for any finite information flow of the insider. In contrast to Kyle (1985) with normal distributions, equilibria exist in general only in mixed strategies and not in pure strategies. In the single-period model we establish bounds for the insider’s strategy in equilibrium. Finally, we prove the existence of an equilibrium for the game with a continuum of actions, by considering an approximating sequence of games with finitely many actions. Because of the lack of compactness of the set of measurable price functions, standard infinite-dimensional fixed point theorems are not applicable.

Self-employment survival in China: does family involvement matter?

Using data from the China Household Finance Survey spanning 2015–2019, we investigate the relationship between family involvement and self-employment survival. Instrumental variables are used to address the potential endogeneity of family involvement. We show that family involvement is crucial in reducing self-employment exit. This significant effect is more pronounced in elderly self-employed and self-employed with children aged 0–5. The underlying mechanisms reveal that family members’ participation in self-employment flexibly allocates idle household labor and improves the household employment rate. Meanwhile, self-employment with family involvement may have better access to credit and better performance, thus enhancing self-employment survival. Our findings are validated by robustness checks, including a discrete time complementary log–log model and estimations using alternative measurements of family involvement and self-employment exit. Our study has useful implications for policy-makers to stimulate family involvement to promote self-employment survival.

Computational approaches on integrating vaccination and treatment strategies in the SIR model using Galerkin time discretization scheme

ABSTRACT In this manuscript, we carried out a thorough analysis of the general SIR model for epidemics. We broadened the model to include vaccination, treatment, and incidence rate. The vaccination rate is a testament to the alternatives made by individuals when it comes to receiving vaccinations and merging with the community of the recovered. The treatment rate measures how often people who have contracted a disease are able to transition into the recovered category. The continuous Galerkin-Petrov method, specifically the cGP(2)-method, is employed to calculate the numerical solutions of the models. The cGP(2)-method imperative to analyse two unknowns over every time interval. The unknowns can be determined by solving a block system of size (2 × 2). This approach demonstrates a strong level of precision throughout the entire time interval, with an impressive rate of convergence at the discrete time points. In addition, the article investigates the concept of the basic reproduction number ( R 0 ) and explores the intricacies of conducting sensitivity analysis of the developed system.

Stage-wise kinetic analysis of ammonia addition effects on two-stage ignition in dimethyl ether

Abstract Ammonia (NH3) has gained considerable attention as a promising carbon-free hydrogen carrier fuel for internal combustion engines, but its direct use in compression-ignition engines presents challenges, often requiring high-reactivity fuels to ignite the premixed NH3/air mixture and initiate combustion. This study focuses on the ignition process of binary NH3 and dimethyl ether (DME) mixtures, as DME is a carbon-neutral, high-reactivity fuel. A key novelty of this paper is the comparison of the ignition processes of DME and NH3/DME mixtures from a temporal, process-oriented perspective, analyzing chemical kinetics across distinct ignition phases rather than focusing solely on instantaneous reactions at discrete time points. The stage-wise analysis reveals that NH3 has minimal impact on the control mechanism governing the two-stage ignition process of DME. Specifically, DME still largely depends on OH radical proliferation during low-temperature oxidation (LTO), which releases heat as the reaction progresses. As the temperature increases, LTO branching pathways gradually shift to chain-propagation pathways, reducing overall reaction activity. The reactivity and temperature rise rate of the system are then governed by the H2O2 loop mechanism before thermal ignition. However, ammonia significantly extends the ignition delay of DME by competing with OH radicals, which are critical for DME oxidation, thus inhibiting ignition. As the ignition reaction proceeds, ammonia kinetics become more involved. For example, nitrogen-containing species from NH3 oxidation, such as NO, NO2, and NH2, react with CH3OCH2 to form CH3OCHO, reducing the flux through the LTO pathway of DME. While ammonia reaction pathways also produce OH radicals, this occurs at the expense of HO2 and H radicals, leading to H2O2 formation. Overall, these findings demonstrate the substantial impact of ammonia addition on DME ignition, highlighting the need for further research to better understand NH3/DME binary fuel ignition and to optimize the design and operation of NH3/DME dual-fuel engines for improved efficiency and reliability.

Computational analysis of the Covid-19 model using the continuous Galerkin–Petrov scheme

Abstract Epidemiological models feature reliable and valuable insights into the prevention and transmission of life-threatening illnesses. In this study, a novel SIR mathematical model for COVID-19 is formulated and examined. The newly developed model has been thoroughly explored through theoretical analysis and computational methods, specifically the continuous Galerkin–Petrov (cGP) scheme. The next-generation matrix approach was used to calculate the reproduction number ( R 0 ) ({R}_{0}) . Both disease-free equilibrium (DFE) ( E 0 ) ({E}^{0}) and the endemic equilibrium ( E ⁎ ) ({E}^{\ast }) points are derived for the proposed model. The stability analysis of the equilibrium points reveals that ( E 0 ) ({E}^{0}) is locally asymptotically stable when R 0 < 1 {R}_{0}\lt 1 , while E ⁎ {E}^{\ast } is globally asymptotically stable when R 0 > 1 {R}_{0}\gt 1 . We have examined the model’s local stability (LS) and global stability (GS) for endemic equilibrium \text{ } and DFE based on the number ( R 0 ) ({R}_{0}) . To ascertain the dominance of the parameters, we examined the sensitivity of the number ( R 0 ) ({R}_{0}) to parameters and computed sensitivity indices. Additionally, using the fourth-order Runge–Kutta (RK4) and Runge–Kutta–Fehlberg (RK45) techniques implemented in MATLAB, we determined the numerical solutions. Furthermore, the model was solved using the continuous cGP time discretization technique. We implemented a variety of schemes like cGP(2), RK4, and RK45 for the COVID-19 model and presented the numerical and graphical solutions of the model. Furthermore, we compared the results obtained using the above-mentioned schemes and observed that all results overlap with each other. The significant properties of several physical parameters under consideration were discussed. In the end, the computational analysis shows a clear image of the rise and fall in the spread of this disease over time in a specific location.

On constraint-conforming numerical discretizations in constitutive material modeling

AbstractFor the modelling of complex materials, internal variables are usually introduced which characterize the microstructural state. Then, evolution equations describe the change of the internal variables due to varying external loading conditions. These equations can be derived, for instance, on the basis of variational principles. The consideration of characteristic observations, such as the preservation of the volume during a change in the microstructural state, can significantly improve the accuracy of the evolution equations. We present a Hamilton principle that provides a unique way to derive evolution equations that obey holonomic constraints and opens up new possibilities for their algorithmic treatment. This is demonstrated for isochoric finite plasticity and phase transformation based on Backward-Euler time discretization. The models presented are efficient and are characterized by simple implementation compared to the exponential map, for example, without suffering a loss of accuracy due to unfulfilled constraints.

Creating and controlling global Greenberger-Horne-Zeilinger entanglement on quantum processors

Greenberger-Horne-Zeilinger (GHZ) states, also known as two-component Schrödinger cats, play vital roles in the foundation of quantum physics and the potential quantum applications. Enlargement in size and coherent control of GHZ states are both crucial for harnessing entanglement in advanced computational tasks with practical advantages, which unfortunately pose tremendous challenges as GHZ states are vulnerable to noise. Here we propose a general strategy for creating, preserving, and manipulating large-scale GHZ entanglement, and demonstrate a series of experiments underlined by high-fidelity digital quantum circuits. For initialization, we employ a scalable protocol to create genuinely entangled GHZ states with up to 60 qubits, almost doubling the previous size record. For protection, we take a different perspective on discrete time crystals (DTCs), originally for exploring exotic nonequilibrium quantum matters, and embed a GHZ state into the eigenstates of a tailor-made cat scar DTC to extend its lifetime. For manipulation, we switch the DTC eigenstates with in-situ quantum gates to modify the effectiveness of the GHZ protection. Our findings establish a viable path towards coherent operations on large-scale entanglement, and further highlight superconducting processors as a promising platform to explore nonequilibrium quantum matters and emerging applications.

Performance Analysis and Parallel Scalability of Numerical Methods for Fractional-in-Space Diffusion Problems with Adaptive Time Stepping

We study numerical methods and algorithms for time-dependent fractional-in-space diffusion problems. The considered anomalous diffusion is modelled by the fractional Laplacian (−Δ)α, 0<α<1, following the integral definition. Fractional diffusion is non-local, and the finite element method (FEM) discretization in space leads to a dense stiffness matrix. It is well known that numerically solving such non-local boundary value problems is expensive. Difficulties increase significantly when the problem is time-dependent. The aim of the article is to develop computationally efficient methods and algorithms. There are two main features of our approach. Hierarchical semi-separable (HSS) compression is applied for an approximate solution of the arising linear systems. For time discretization, we use an adaptive forward–backward Euler scheme. The properties of the composite algorithm thus obtained are investigated. In particular, the block representation of HSS compression allowed us to upgrade the HSS solver to efficiently handle varying diagonally perturbed transition matrices corresponding to changing time steps. The contribution of the paper is threefold. The methods are completely constructive, which allows for a clearly structured description of the algorithms. A theoretical estimate of the computational complexity is presented. It shows the advantages of the adaptive time stepping in combination with the HSS solver. Theoretical results are supported by representative numerical experiments. Both sequential and parallel scalability and efficiency are analyzed. The presented results provide convincing proof of the concept of the proposed methods and algorithms.