Time-dependent Operator Research Articles

The associations of triglyceride glucose and its derived indexes with 10-year cardiovascular disease risk in a Chinese community-based population

Abstract Background The triglyceride-glucose (TyG) index has been regarded as a reliable surrogate marker of insulin resistance and an independent predictor of cardiovascular outcomes. However, the associations of the derived TyG indexes with incident CVD risk are controversial, and that the relevant data are sparse for the Chinese population studies. Objectives We aimed to investigate the association of TyG index and its derived indexes with incident CVD risk and to compare their predictive capacity in the Chinese community-based population. Methods We conducted a prospective cohort study on subjects from a Beijing community in China who were free of prevalent stroke or myocardial infarction at baseline. The primary endpoint was a new-onset major adverse cardiovascular event (MACE), defined as a composite of cardiovascular death, first non-fatal stroke, and first non-fatal acute myocardial infarction. Cox proportional-hazards regression analyses and time-dependent receiver operator characteristic curve analyses were performed. Results A total of 8,806 participants were included. The mean age was 56.51 ± 8.93 years, and 5,679 (64.5%) were female. The median value of TyG index, TyG-BMI, TyG-WC, and TyG-WHtR were 8.71 (8.34-9.12), 226.37 (201.02-252.87), 720.80 (653.52-791.03), and 4.47 (4.07-4.90), respectively. During a mean 9.7-year follow-up, a total of 877 (10%) cases of MACE were recorded. The Kaplan–Meier curves showed significant differences in cumulative hazards of new-onset MACE among tertiles of abovementioned indexes (Log rank P < 0.0001) (Figure 1). Multivariable Cox regression analyses revealed that TyG index, TyG-BMI, TyG-WC, and TyG-WHtR were positively associated with the hazards of MACE. Compared with tertile 1, the multivariable adjusted HRs (95% CIs) of incident CVD for tertile 3 were 1.26 (1.01-1.56), 1.57 (1.15-2.14), 1.54 (1.19-2.00), and 1.81 (1.40-2.33) for TyG, TyG-BMI, TyG-WC, and TyG-WHtR, respectively (Figure 2). There are no significant differences in the Harrell's C indexes among the various indexes compared to the TyG index (TyG index: 0.758 [0.743-0.774], TyG-BMI: 0.758 [0.743-0.773], TyG-WC: 0.758 [0.743-0.774], and TyG-WHtR: 0.760 [0.745-0.775]). Conclusions Higher TyG and its derived indexes are significantly associated with an increased risk of 10-year MACE. Our results suggest that TyG and its derived indexes may serve as a useful marker for stratifying CVD risk in the general population.

Inverted oscillator: quantum discrete spectrum

Abstract We use the invariant operator method to investigate the quantum characteristics of the inverted oscillator. We introduce a unitary transformation that maps a time-dependent general Hermitian linear invariant operator to a time-independent one, which describes a harmonic oscillator with a unit mass and a constant frequency. Our investigation includes three distinguished cases: negative frequency, zero frequency, and positive frequency. Our most interesting result concerns the last case. For the case of negative frequency, we propose a new inner product by the introduction of the new metric operator η. Coherent state for this case will be constructed.

Assessing and predicting type 2 diabetes risk with triglyceride glucose-body mass index in the Chinese nondiabetic population-Data from long-term follow-up of Da Qing IGT and Diabetes Study.

We intended to characterize the superiority of triglyceride glucose-body mass index (TyG-BMI) in predicting type 2 diabetes mellitus (T2DM) compared with triglyceride glucose (TyG) and homeostatic model assessment for insulin resistance (HOMA-IR). A total of 699 nondiabetic participants in the Da Qing IGT and Diabetes Study were involved in the present analysis and classified according to the median of baseline TyG-BMI, namely the G1 (low TyG-BMI) and G2 (high TyG-BMI) groups. Information on developing diabetes was assessed from 1986 to 2020. During the 34-year follow-up, after adjustment for confounders, the G2 group had a higher risk of developing type 2 diabetes than the G1 group (hazard ratio [HR]: 1.92, 95% confidence interval [CI]: 1.51-2.45, p < 0.0001). Restricted cubic spline analyses showed that increased TyG-BMI was linearly related to higher risks of type 2 diabetes (p for non-linearity>0.05). Time-dependent receiver operator characteristics curves suggested that TyG-BMI exhibited higher predictive ability than TyG (6-year: area under the curve [AUC]TyG-BMI vs. AUCTyG, 0.78 vs. 0.70, p = 0.03; 34-year: AUCTyG-BMI vs. AUCTyG, 0.79 vs. 0.73, p = 0.04) and HOMA-IR (6-year: AUCTyG-BMI vs. AUCHOMA-IR, 0.78 vs. 0.70, p = 0.07; 34-year: AUCTyG-BMI vs. AUCHOMA-IR, 0.79 vs. 0.71, p = 0.04) in both short and long terms, and the thresholds of TyG-BMI to predict type 2 diabetes were relatively stable (195.24-208.41) over the 34-year follow-up. In this post hoc study, higher TyG-BMI was associated with an increased risk of type 2 diabetes and demonstrated better predictability than TyG and HOMA-IR, favoring the application of TyG-BMI as a potential tool for evaluating the risk of type 2 diabetes in clinical practice.

Error Estimates for First- and Second-Order Lagrange–Galerkin Moving Mesh Schemes for the One-Dimensional Convection–Diffusion Equation

A new moving mesh scheme based on the Lagrange–Galerkin method for the approximation of the one-dimensional convection–diffusion equation is studied. The mesh movement is prescribed by a discretized dynamical system for the nodal points. This system is related to the velocity and diffusion coefficient in the convection–diffusion equation such that the nodal points follow the convective flow of the model. It is shown that under a restriction of the time step size the mesh movement cannot lead to an overlap of the elements and therefore an invalid mesh. Using a piecewise linear approximation, optimal error estimates in the ℓ∞(L2)∩ℓ2(H01)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^\\infty (L^2) \\cap \\ell ^2(H_0^1)$$\\end{document} norm are proved in case of both, a first-order backward Euler method and a second-order two-step method in time. These results are based on new estimates of the time dependent interpolation operator derived in this work. Preservation of the total mass is verified for both choices of the time discretization. Numerical experiments are presented that confirm the error estimates and demonstrate that the proposed moving mesh scheme can circumvent limitations that the Lagrange–Galerkin method on a fixed mesh exhibits.

The application of lung immune prognostic index in predicting the prognosis of 302 STS patients.

Soft tissue sarcoma (STS) are heterogeneous and rare tumors, and few studies have explored predicting the prognosis of patients with STS. The Lung Immune Prognostic Index (LIPI), calculated based on baseline serum lactate dehydrogenase (LDH) and the derived neutrophils/(leukocytes minus neutrophils) ratio (dNLR), was considered effective in predicting the prognosis of patients with pulmonary cancer and other malignancies. However, the efficacy of the LIPI in predicting the prognosis of patients with STS remains unclear. This study retrospectively reviewed patients with STS admitted to our center from January 2016 to January 2021. Their hematological and clinical characteristics were collected and analyzed to construct the LIPI specific to STS. The correlations between various predictive factors and overall survival (OS) were examined using Kaplan-Meier and Cox regression analyses. Independent risk factors for OS were identified using univariate and multivariate analyses. Finally, a LIPI nomogram model for STS was established. This study enrolled 302 patients with STS, of which 87 (28.9%), 162 (53.6%), and 53 (17.5%) were classified into three LIPI-based categories: good, moderate, and poor, respectively (P < 0.0001). The time-dependent operator curve showed that the LIPI had better prognostic predictive ability than other hematological and clinical characteristics. Univariate and multivariate analyses identified the Fédération Nationale des Centres de Lutte Contre le Cancer grade (FNCLCC/G), tumor size, and LIPI as independent risk factors. Finally, a nomogram was constructed by integrating the significant prognostic factors. Its C-index was 0.72, and the calibration curve indicated that it could accurately predict the three- and five-year OS of patients with STS. The decision and clinical impact curves also indicated that implementing this LIPI-nomogram could significantly benefit patients with STS. This study explored the efficacy of the LIPI in predicting the prognosis of 302 patients with STS, classifying them into three categories to evaluate the prognosis. It also reconstructed a LIPI-based nomogram to assist clinicians in predicting the three- and five-year OS of patients with STS, potentially enabling timely intervention and customized management.

Heat equation on the hypergraph containing vertices with given data

This paper is concerned with a multivalued ordinary differential equation governed by the hypergraph Laplacian, which describes the diffusion of “heat” or “particles” on the vertices of hypergraph. We consider the case where the heat on several vertices is manipulated internally by the observer, namely, are fixed by some given functions. This situation can be reduced to a nonlinear evolution equation associated with a time-dependent subdifferential operator, whose solvability has been investigated in numerous previous researches. In this paper, we give an alternative proof of the solvability in order to avoid some complicated calculations arising from the chain rule for the time-dependent subdifferential. As for results which cannot be assured by the known abstract theory, we also discuss the continuous dependence of solution on the given data and the time-global behavior of solution.

On high-order schemes for the space-fractional conservative Allen–Cahn equations with local and local–nonlocal operators

In this study, we focus on two fractional conservative Allen–Cahn equations with a nonlocal space-independent operator (called the RSLM operator) and a local–nonlocal space–time dependent operator (called the BBLM operator), respectively. Recently, scholars have found that the fractional Allen–Cahn equation is better than the classical equation for describing the interface thickness. Subsequently, the fractional conservative Allen–Cahn equation was used for conservative fluid dynamics. We find that two fractional conservative Allen–Cahn equations both hold the mass conservation, energy stability, and maximum bound principle. In this study, we construct second- and third-order numerical algorithms for two fractional conservative Allen–Cahn equations, and prove the mass conservation and energy decrease laws for them. To our knowledge, for the conservative Allen–Cahn equation, there are only a few works for high-order schemes with proof of their energy stability. We then present several numerical examples whose results show that the convergence orders of the numerical solutions reach the second- and third-order in time as expected. We also simulate some phase transition behaviors to be mass conservative and energy stable, as described by the fractional conservative Allen–Cahn equations, and have some interesting discoveries. It is found that the BBLM formulation can better maintain small features, but the RSLM formulation is better at capturing the profile of the interface. We also compare two fractional conservation equations with the fractional Cahn–Hilliard equation, whose changes are similar to those of the RSLM formulation. In particular, we propose the variable-step BDF2 and BDF3 methods, and prove the mass conservation and energy stability under a time-ratio constraint.

On a Non-Convex Lagrange Optimal Control Problem

Abstract In this paper, we are concerned with an iterative differential inclusion governed by the time-dependent maximal monotone operator with perturbation. The approach to solve our problem is based on the Yosida approximation technique. The theoretical result is applied to prove an existence result for a Lagrange optimal control problem without assumptions concerning convexity.

Metabolic score for insulin resistance predicts major adverse cardiovascular event in premature coronary artery disease.

The Metabolic Score for Insulin Resistance (METS-IR) index serves as a simple surrogate marker for insulin resistance (IR) and is associated with the presence and severity of coronary artery disease (CAD). However, the prognostic significance of METS-IR in patients with premature CAD remains unclear. This study aims to investigate the prognostic value of METS-IR in premature CAD. This retrospective study included 582 patients diagnosed with premature CAD between December 2012 and July 2019. The median follow-up duration was 63 months (interquartile range, 44-81 months). The primary endpoint was Major Adverse Cardiovascular Events (MACE), defined as a composite of all-cause death, non-fatal myocardial infarction (MI), repeat coronary artery revascularization, and non-fatal stroke. Patients with MACE had significantly higher METS-IR levels than those without MACE (44.88±8.11 vs. 41.68±6.87, p<0.001). Kaplan-Meier survival curves based on METS-IR tertiles demonstrated a statistically significant difference (log-rank test, p<0.001). In the fully adjusted model, the Hazard Ratio (95% CI) for MACE was 1.41 (1.16-1.72) per SD increase in METS-IR, and the P for trend based on METS-IR tertiles was 0.001 for MACE. Time-dependent Receiver Operator Characteristic (ROC) analysis of METS-IR yielded an Area Under the Curve (AUC) of 0.74 at 2 years, 0.69 at 4 years, and 0.63 at 6 years. METS-IR serves as a reliable prognostic predictor of MACE in patients with premature CAD. Therefore, METS-IR may be considered a novel, cost-effective, and dependable indicator for risk stratification and early intervention in premature CAD.

Perturbation of parabolic equations with time-dependent linear operators: convergence of linear processes and solutions

Perturbation of parabolic equations with time-dependent linear operators: convergence of linear processes and solutions

Data-driven models of nonautonomous systems

Nonautonomous dynamical systems are characterized by time-dependent inputs, which complicates the discovery of predictive models describing the spatiotemporal evolution of the state variables of quantities of interest from their temporal snapshots. When dynamic mode decomposition (DMD) is used to infer a linear model, this difficulty manifests itself in the need to approximate the time-dependent Koopman operators. Our approach is to approximate the original nonautonomous system with a modified system derived via a local parameterization of the time-dependent inputs. The modified system comprises a sequence of local parametric systems, which are subsequently approximated by a parametric surrogate model using the DRIPS (dimension reduction and interpolation in parameter space) framework. The offline step of DRIPS relies on DMD to build a linear surrogate model, endowed with reduced-order bases for the observables mapped from training data. The online step interpolates on suitable manifolds to construct a sequence of iterative parametric surrogate models; the target/test parameter points on these manifolds are specified by a local parameterization of the test time-dependent inputs. We use numerical experimentation to demonstrate the robustness of our method and compare its performance with that of deep neural networks.

Periodic systems with time dependent maximal monotone operators

Periodic systems with time dependent maximal monotone operators

BV Solutions to Evolution Inclusion with a Time and Space Dependent Maximal Monotone Operator

This paper deals with the research of solutions of bounded variation (BV) to evolution inclusion coupled with a time and state dependent maximal monotone operator. Different problems are studied: existence of solutions, unicity of the solution, existence of periodic and bounded variation right continuous (BVRC) solutions. Second-order evolution inclusions and fractional (Caputo and Riemann–Liouville) differential inclusions are also considered. A result of the Skorohod problem driven by a time- and space-dependent operator under rough signal and a Volterra integral perturbation in the BRC setting is given. The paper finishes with some results for fractional differential inclusions under rough signals and Young integrals. Many of the given results are novel.

Invariant Closed Sets with Respect to Differential Inclusions with Time-Dependent Maximal Monotone Operators

Invariant Closed Sets with Respect to Differential Inclusions with Time-Dependent Maximal Monotone Operators

Microscopic theory of a Janus motor in a non-equilibrium fluid: Surface hydrodynamics and boundary conditions.

We present a derivation from the first principles of the coupled equations of motion of an active self-diffusiophoretic Janus motor and the hydrodynamic densities of its fluid environment that are nonlinearly displaced from equilibrium. The derivation makes use of time-dependent projection operator techniques defined in terms of slowly varying coarse-grained microscopic densities of the fluid species number, total momentum, and energy. The exact equations of motion are simplified using time scale arguments, resulting in Markovian equations for the Janus motor linear and angular velocities with average forces and torques that depend on the fluid densities. For a large colloid, the fluid equations are separated into bulk and interfacial contributions, and the conditions under which the dynamics of the fluid densities can be accurately represented by bulk hydrodynamic equations subject to boundary conditions on the colloid are determined. We show how the results for boundary conditions based on continuum theory can be obtained from the molecular description and provide Green-Kubo expressions for all transport coefficients, including the diffusiophoretic coupling and the slip coefficient.

Nonlinear generalized master equations: quantum case

A system of N≫1 interacting spinless quantum particles, described by a statistical operator F(t), is considered. A time-dependent projection operator formalism for a family of projectors, which select a statistical operator FS(t) for a group of S < N relevant particles by integration of the variables of the irrelevant N − S ‘environment’ particles, is presented. This formalism results in a nonlinear version of the inhomogeneous Nakajima–Zwanzig generalized master equation (GME) for the relevant part of the statistical operator F(t), which contains an undesirable initial condition term related to the irrelevant part of the statistical operator depending on N variables. By introducing an additional operator identity, the obtained nonlinear inhomogeneous Nakajima–Zwanzig equation is exactly converted into a homogeneous nonlinear equation, accounting for initial correlations in the kernel governing its evolution. Both of these equations are equivalent to the corresponding evolution equations for FS(t) and take into account the dynamics of the environment particles. The equations for FS(t) are specialized in the linear approximation for the particle density n and analyzed for a one-particle statistical operator F1(t) . It turns out that, in addition to the correlations caused by interparticle interaction, the irrelevant initial condition term in the inhomogeneous equation for F1(t) , defined by the two-particle correlations, also contains quantum correlations conditioned by the quantum particle statistics. The latter do not vanish with time and cannot be disregarded in the conventional way by applying e.g. Bogoliubov’s principle of weakening of initial correlations. In order to take initial quantum and other correlations (commonly discarded in an unconvincing way) into consideration, the obtained nonlinear homogeneous GME has been applied. Exact in the first approximation in n, we obtain a new homogeneous equation for F1(t) , which accounts for correlations and is valid for any time moment t. It is shown how this reversible-in-time equation changes at different timescales and converts into the irreversible quantum Boltzmann equation on the macroscopic timescale without the use of the conventional approximations.

Prognostic Value of Nighttime Double Product in Nondialysis Chronic Kidney Disease With Hypertension.

Both nighttime systolic blood pressure and pulse rate are associated with adverse outcomes in patients with chronic kidney disease (CKD). However, nighttime double product (DP), which is the product of nighttime systolic blood pressure and pulse rate, has not yet been investigated in this context. The present study aimed to explore the prognostic value of nighttime DP for adverse outcomes in patients with CKD and hypertension. This retrospective cohort study included a total of 1434 patients with nondialysis CKD complicated by hypertension. The patients were enrolled in Zhuhai and Guangzhou, China, with a median follow-up of 23.8 months. Patient enrollment for the high or low nighttime DP group was performed on the basis of the cutoff value determined by time-dependent receiver operator characteristic curve analysis. The primary end point was a composite of major cardiovascular and cerebrovascular events, and the secondary end point was all-cause death and composite renal end point. The 24-hour circadian DP rhythm was established via multiple-component cosinor analysis. Cox regression was used to explore the association between nighttime DP and adverse outcomes. The DP of nondialysis patients with CKD and hypertension showed a diurnal rhythm, which varied with renal function. After adjustment, high nighttime DP was associated with a higher risk for major cardiovascular and cerebrovascular events (hazard ratio [HR], 5.823 [95% CI, 2.382-14.233]), all-cause death (HR, 4.978 [95% CI, 2.205-11.240]), and composite renal event (HR, 1.661 [95% CI, 1.128-2.447]), compared with low nighttime DP. These associations were independent of nighttime systolic blood pressure and PR. The present cohort study demonstrated that DP had diurnal fluctuations and nighttime DP was an important prognostic factor in nondialysis patients with CKD and hypertension, outperforming traditional risk factors, including systolic blood pressure and pulse rate.

Long-time behavior of nonclassical diffusion equations with memory on time-dependent spaces

This paper aims to study the long-time behavior of nonclassical diffusion equation with memory and disturbance parameters on time-dependent space. By using the contractive process method on the family of time-dependent spaces and operator decomposition technique, the existence of pullback attractors is first proved. Then the upper semi-continuity of pullback attractors with respect to perturbation parameter ν in M t is obtained. It’s remarkable that the nonlinearity f satisfies the polynomial growth of arbitrary order.

Reducibility of a class of operators induced by the dispersive third order Benjamin-Ono equation

We prove the reducibility of a class of quasi-periodically time dependent linear operators, which are derived from linearizing the dispersive third order Benjamin–Ono (BO) equation on the circle at a small amplitude quasi-periodic function, with a diophantine frequency vector ω∈O0⊂Rν. It is shown that there exists a set O∞⊂O0 of asymptotically full Lebesgue measure such that for any ω∈O∞, the operators can be reduced to the ones with constant coefficients by some linear transformations depending on time quasi-periodically. These transformations include a change of variable induced by a diffeomorphism of the torus, the flow of some partial differential equations and a pseudo-differential operator of order zero. We first reduce the linearized operator of order three to the one with constant coefficients plus a remainder of order zero, and then a perturbative reducibility scheme is performed. The major difficulties encountered are brought by the non-smooth character of the dispersive relation in view of the presence of the Hilbert operator H. We look for several appropriate transformations which are real, reversibility-preserving and satisfy the sharp tame bounds which are used for the reducibility. This work will be the first fundamental step in proving the existence of time quasi-periodic solutions for the dispersive third order BO equation.

Construction and validation of a nomogram to predict the overall survival of small cell lung cancer: a multicenter retrospective study in Shandong province, China

BackgroundPatients diagnosed with small cell lung cancer (SCLC) typically experience a poor prognosis, and it is essential to predict overall survival (OS) and stratify patients based on distinct prognostic risks.MethodsTotally 2309 SCLC patients from the hospitals in 15 cities of Shandong from 2010 − 2014 were included in this multicenter, population-based retrospective study. The data of SCLC patients during 2010–2013 and in 2014 SCLC were used for model development and validation, respectively. OS served as the primary outcome. Univariate and multivariate Cox regression were applied to identify the independent prognostic factors of SCLC, and a prognostic model was developed based on these factors. The discrimination and calibration of this model were assessed by the time-dependent C-index, time-dependent receiver operator characteristic curves (ROC), and calibration curves. Additionally, Decision Curve Analysis (DCA) curves, Net Reclassification Improvement (NRI), and Integrated Discriminant Improvement (IDI) were used to assess the enhanced clinical utility and predictive accuracy of the model compared to TNM staging systems.ResultsMultivariate analysis showed that region (Southern/Eastern, hazard ratio [HR] = 1.305 [1.046 − 1.629]; Western/Eastern, HR = 0.727 [0.617 − 0.856]; Northern/Eastern, HR = 0.927 [0.800 − 1.074]), sex (female/male, HR = 0.838 [0.737 − 0.952]), age (46–60/≤45, HR = 1.401 [1.104 − 1.778]; 61–75/≤45, HR = 1.500 [1.182 − 1.902]; >75/≤45, HR = 1.869 [1.382 − 2.523]), TNM stage (II/I, HR = 1.119[0.800 − 1.565]; III/I, HR = 1.478 [1.100 − 1.985]; IV/I, HR = 1.986 [1.477 − 2.670], surgery (yes/no, HR = 0.677 [0.521 − 0.881]), chemotherapy (yes/no, HR = 0.708 [0.616 − 0.813]), and radiotherapy (yes/no, HR = 0.802 [0.702 − 0.917]) were independent prognostic factors of SCLC patients and were included in the nomogram. The time-dependent AUCs of this model in the training set were 0.699, 0.683, and 0.683 for predicting 1-, 3-, and 5-year OS, and 0.698, 0.698, and 0.639 in the validation set, respectively. The predicted calibration curves aligned with the ideal curves, and the DCA curves, the IDI, and the NRI collectively demonstrated that the prognostic model had a superior net benefit than the TNM staging system.ConclusionThe nomogram using SCLC patients in Shandong surpassed the TNM staging system in survival prediction accuracy and enabled the stratification of patients with distinct prognostic risks based on nomogram scores.