Endpoints Of Interval Research Articles

On an Exact Convergence of Quasi-Periodic Interpolations for the Polyharmonic–Neumann Eigenfunctions

Fourier expansions employing polyharmonic–Neumann eigenfunctions have demonstrated improved convergence over those using the classical trigonometric system, due to the rapid decay of their Fourier coefficients. Building on this insight, we investigate interpolations on a finite interval that are exact for polyharmonic–Neumann eigenfunctions and exhibit similar benefits. Furthermore, we enhance the convergence of these interpolations by incorporating the concept of quasi-periodicity, wherein the basis functions are periodic over a slightly extended interval. We demonstrate that those interpolations achieve significantly better convergence rates away from the endpoints of the approximation interval and offer increased accuracy over the entire interval. We establish these properties for a specific case of polyharmonic–Neumann eigenfunctions known as the modified Fourier system. For other basis functions, we provide supporting evidence through numerical experiments. While the latter methods display superior convergence rates, we demonstrate that interpolations using the modified Fourier basis offer distinct advantages. Firstly, they permit explicit representations via the inverses of certain Vandermonde matrices, whereas other interpolation methods require approximate computations of the eigenvalues and eigenfunctions involved. Secondly, these matrix inverses can be efficiently computed for numerical applications. Thirdly, the introduction of quasi-periodicity improves the convergence rates, making them comparable to those of other interpolation techniques.

Inverse coefficient problems for one-dimensional time-fractional diffusion equations

We prove the uniqueness in determining a spatially varying zeroth-order coefficient of a one-dimensional time-fractional diffusion equation by initial value and Cauchy data at one end point of the spatial interval.

Generalised knotoids

Abstract In 2010, Turaev introduced knotoids as a variation on knots that replaces the embedding of a circle with the embedding of a closed interval with two endpoints which here we call poles. We define generalised knotoids to allow arbitrarily many poles, intervals and circles, each pole corresponding to any number of interval endpoints, including zero. This theory subsumes a variety of other related topological objects and introduces some particularly interesting new cases. We explore various analogs of knotoid invariants, including height, index polynomials, bracket polynomials and hyperbolicity. We further generalise to knotoidal graphs, which are a natural extension of spatial graphs that allow both poles and vertices.

Inverse Problems for One-Dimensional Fluid-Solid Interaction Models

AbstractWe consider a one-dimensional fluid-solid interaction model governed by the Burgers equation with a time varying interface. We discuss the inverse problem of determining the shape of the interface from Dirichlet and Neumann data at one endpoint of the spatial interval. In particular, we establish unique results and some conditional stability estimates. For the proofs, we use and adapt some lateral estimates that, in turn, rely on appropriate Carleman and interpolation inequalities.

Interval-Censored Linear Quantile Regression

Censored quantile regression has emerged as a prominent alternative to classical Cox’s proportional hazards model or accelerated failure time model in both theoretical and applied statistics. While quantile regression has been extensively studied for right-censored survival data, methodologies for analyzing interval-censored data remain limited in the survival analysis literature. This article introduces a novel local weighting approach for estimating linear censored quantile regression, specifically tailored to handle diverse forms of interval-censored survival data. The estimation equation and the corresponding convex objective function for the regression parameter can be constructed as a weighted average of quantile loss contributions at two interval endpoints. The weighting components are nonparametrically estimated using local kernel smoothing or ensemble machine learning techniques. To estimate the nonparametric distribution mass for interval-censored data, a modified EM algorithm for nonparametric maximum likelihood estimation is employed by introducing subject-specific latent Poisson variables. The proposed method’s empirical performance is demonstrated through extensive simulation studies and real data analyses of two HIV/AIDS datasets. Supplementary materials for this article are available online.

Cardiovascular and Cerebrovascular Adverse Events Associated with Intravitreal Anti-VEGF Monoclonal Antibodies: A World Health Organization Pharmacovigilance Study

PurposeTo analyze cardiovascular and cerebrovascular adverse events (ADRs) after intravitreal anti-vascular endothelial growth factor (VEGF; aflibercept, bevacizumab, brolucizumab, and ranibizumab) treatment. SubjectsVigiBase, a World Health Organization (WHO) global safety report database DesignPharmacovigilance study MethodsThe individual-case-safety reports (ICSR) of cardiovascular and cerebrovascular ADRs after intravitreal anti-VEGF treatment were compared with those reported in the full database. From 2004 to 2023, 23,129 ADRs after intravitreal anti-VEGF therapy and 25,015,132 ADRs associated with any drug (full database). Main Outcome MeasuresThe reporting odds ratio (ROR) and information components (IC) were calculated, and the 95% lower credibility interval endpoint of the information component (IC025) was used for disproportionate Bayesian reporting. Inter-drug comparisons were performed using the ratio of odd ratio (rOR). ResultsCompared with the full database, anti-VEGFs were associated with an increased reporting of myocardial infarction (IC025 0.75; ROR: 1.78 [95% CI 1.70–1.86]), angina pectoris (IC025 0.53; ROR: 1.61 [95% CI 1.47–1.77]), arrythemias including atrial fibrillation, atrial flutter, ventricular fibrillation, supraventricular tachycardia (all IC025 >0, ROR>1), hypertension (IC025 2.22; ROR: 4.91 [95% CI 4.82–5.01]), and hypertensive crisis (IC025 1.97; ROR: 4.49 [95% CI 4.07–4.97]). Moreover, anti-VEGFs were associated with a higher reporting of cerebrovascular ADRs such as cerebral infarction (IC025 4.34; ROR: 23.19 [95% CI 22.10–24.34]), carotid artery stenosis (IC025 1.85; ROR: 5.24 [95% CI 3.98–6.89]), cerebral hemorrhage (IC025 2.29; ROR: 5.38 [95% CI 5.03–5.76]), and subarachnoid hemorrhage (IC025 1.98; ROR: 4.81 [95% CI 4.14–5.6]). Inter-drug comparison indicated that compared to ranibizumab, patients with aflibercept showed overall under-reporting of cardiovascular and cerebrovascular ADRs such as myocardial infarction (rOR 0.55 [95% CI 0.49–0.52]), atrial fibrillation (rOR 0.28 [95% CI 0.23–0.35]), cerebrovascular accident (rOR, 0.15 [95% CI 0.14–0.17]), and cerebral hemorrhage (rOR, 0.51 [95% CI 0.40–0.65]). ConclusionsIn this pharmacovigilance case-noncase study, significantly increased reporting of cardiovascular and cerebrovascular ADRs were identified after intravitreal anti-VEGF treatment. While ranibizumab may exhibit superior systemic safety regarding its biological characteristics, it is crucial not to overlook the occurrence of cardiovascular and cerebrovascular ADRs considering its higher reporting rate than bevacizumab or aflibercept.

Semiparametric g-computation for survival outcomes with time-fixed exposures: An illustration

PurposeGeneralized (g-) computation is a useful tool for causal inference in epidemiology. However, in settings when the outcome is a survival time subject to right censoring, the standard pooled logistic regression approach to g-computation requires arbitrary discretization of time, parametric modeling of the baseline hazard function, and the need to expand one’s dataset. We illustrate a semiparametric Breslow estimator for g-computation with time-fixed treatments and survival outcomes that is not subject to these limitations. MethodsWe compare performance of the Breslow g-computation estimator to the pooled logistic g-computation estimator in simulations and illustrate both approaches to estimate the effect of a 3-drug vs 2-drug antiretroviral therapy regimen among people with HIV. ResultsIn simulations, both approaches performed well at the end of follow-up. The pooled logistic approach was biased at times between the endpoints of the discrete time intervals used, while the Breslow approach was not. In the example, both approaches estimated a 1-year risk difference of about 6 % in favor of the 3-drug regimen, but the shape of the survival curves differed. ConclusionsThe Breslow g-computation estimator of counterfactual risk functions does not rely on strong parametric assumptions about the time-to-event distribution or onerous dataset expansions.

Generalized log orthogonal functions spectral collocation method for two dimensional weakly singular Volterra integral equations of the second kind

AbstractIn this article, a generalized log orthogonal functions (GLOFs)‐spectral collocation method to two dimensional weakly singular Volterra integral equations of the second kind is proposed. The mild singularities of the solution at the interval endpoint can be captured by Gauss‐GLOFs quadrature and the shortcoming of the traditional spectral method which cannot well deal with weakly singular Volterra integral equations with limited regular solutions is avoided. A detailed convergence analysis of the numerical solution is carried out. The efficiency of the proposed method is demonstrated by numerical examples.

An efficient numerical method based on ultraspherical polynomials for linear weakly singular fractional Volterra integro‐differential equations

The linear weakly singular fractional Volterra integro‐differential equations involving the Caputo derivative have solutions whose derivatives are unbounded at the left endpoint of the integration interval. In this paper, we use suitable transformations to prevail on this nonsmooth behavior. We used the product integration method based on the new fractional basis function to solve these equations, which led to the production of the new interpolation formula and weights for the method. We investigate the convergence of the presented method. The proposed scheme is employed to solve some numerical examples to test its efficiency and accuracy.

Exact simulation of the Hull and White stochastic volatility model

We show how to simulate exactly the asset price and the variance under the Hull and White stochastic volatility model. We derive analytical formulas for the Laplace transform of the time integral of volatility conditional on the variance level at the endpoint of the time interval and the Laplace transform of integrated variance conditional on both integrated volatility and variance. Based on these results, we simulate the model through a nested-conditional factorization approach, where Laplace transforms are inverted through the (conditional) Fourier-cosine (COS) method. Under this model, our approach can be used to generate unbiased estimates for the price of derivatives instruments. We propose some variants of the exact simulation scheme for computing unbiased estimates of option prices and sensitivities, a difficult task in the Hull and White model. These variants also allow for a significant reduction in the Monte Carlo simulation estimator's variance (around 93-98%) and the computing time (around 22%) when pricing options. The performances of the proposed algorithms are compared with various benchmarks. Numerical results demonstrate the faster convergence rate of the error in our method, which achieves an O(s−1/2) convergence rate, where s is the total computational budget, largely outperforming the benchmark.

HETerogeneous vectorized or Parallel (HETPv1.0): an updated inorganic heterogeneous chemistry solver for the metastable-state NH4+–Na+–Ca2+–K+–Mg2+–SO42−–NO3−–Cl−–H2O system based on ISORROPIA II

Abstract. We describe a new Fortran computer program to solve the system of equations for the NH4+–Na+–Ca2+–K+–Mg2+–SO42-–NO3-–Cl−–H2O system, based on the algorithms of ISORROPIA II. Specifically, the code solves the system of equations describing the “forward” (gas + aerosol input) metastable state but with algorithm improvements and corrections. These algorithm changes allow the code to deliver more accurate solution results in formal evaluations of accuracy of the roots of the systems of equations, while reducing processing time in practical applications by about 50 %. The improved solution performance results from several implementation improvements relative to the original ISORROPIA algorithms. These improvements include (i) the use of the “interpolate, truncate and project” (ITP) root-finding approach rather than bisection, (ii) the allowance of search interval endpoints as valid roots at the onset of a search, (iii) the use of a more accurate method to solve polynomial subsystems of equations, (iv) the elimination of negative concentrations during iterative solutions, (v) corrections for mass conservation enforcement, and (vi) several code structure improvements. The new code may be run in either a “vectorization” mode wherein a global convergence criterion is used across multiple tests within the same chemical subspace or a “by case-by-case” mode wherein individual test cases are solved with the same convergence criteria. The latter approach was found to be more efficient on the compiler tested here, but users of the code are recommended to test both options on their own systems. The new code has been constructed to explicitly conserve the input mass for all species considered in the solver and is provided as open-source Fortran shareware.

Segmental estimation and testing method for power-law distributions and some extensions

For discrete segmental power-law distributions, the probability ratio ft/ft+1 is a linear function of the exponent parameter. Based on this property, the estimation of the exponent parameter and a goodness-of-fit testing method are provided. The proposed testing method is parameter-independent and the testing statistic is proved to asymptotically follow a chi-square distribution. In the region where power-law properties exist, the testing method and the estimation method can be applied by segments, so they can also be used to determine interval endpoints. These methods can also be extended to other discrete distributions such as Yule distribution, Poisson distribution, Geometry distribution and so on. Some simulation results of synthetic truncated power-law distributions provide support for the effectiveness of the proposed methods. To demonstrate the applicability of the method, two empirical examples, word frequencies of the novel Moby Dick and US casuality numbers in the American Indian War, are analyzed.

Long-Term Follow-Up of the Anthracyclines in Early Breast Cancer Trials (USOR 06-090, NSABP B-46-I/USOR 07132, and NSABP B-49 [NRG Oncology]).

Clinical trials frequently include multiple end points that mature at different times. The initial report, typically based on the primary end point, may be published when key planned co-primary or secondary analyses are not yet available. Clinical Trial Updates provide an opportunity to disseminate additional results from studies, published in JCO or elsewhere, for which the primary end point has already been reported.The primary joint efficacy analysis of the Anthracyclines in Early Breast Cancer (ABC) trials reported in 2017 failed to demonstrate nonanthracycline adjuvant therapy was noninferior to anthracycline-based regimens in high-risk, early breast cancer. Full analyses of the studies had proceeded when the prespecified futility boundary was crossed at a planned futility analysis for the ability to demonstrate noninferiority of a nonanthracycline regimen with continued follow-up. These results were presented with 3.3 years of median follow-up. This manuscript reports results of the final analyses of the study efficacy end points conducted with 6.9 years of median follow-up. Long-term analysis of invasive disease-free survival (IDFS), the primary end point of the ABC trials, remains consistent with the original results, as noninferiority of the nonanthracycline regimens could not be declared on the basis of the original criteria. The secondary end point of recurrence-free interval, which excluded deaths not due to breast cancer as events, favored anthracycline-based regimens, and tests for heterogeneity were significant for hormone receptor status (P = .02) favoring anthracycline regimens for the hormone receptor-negative cohorts. There was no difference in overall survival, and review of the type of IDFS events in the groups suggested reductions in cancer recurrences achieved with anthracycline regimens were offset by late leukemias and deaths unrelated to breast cancer.

A fundamental sequences method with time-reduction for one-dimensional lateral Cauchy problems

A fundamental sequences method is derived for the numerical solution of an ill-posed one-dimensional lateral Cauchy problem for a hyperbolic damped wave equation, including as a special case the parabolic heat equation. Either the Laguerre transform or the Houbolt finite difference scheme is applied to reduce the time-dependent lateral Cauchy problem to a sequence of second-order ordinary differential equations (ODEs) with function values and derivatives specified at the right endpoint of a finite space interval. A set of fundamental solutions is constructed, termed a fundamental sequence, to the differential equations. The solution of the obtained ODEs is approximated by a linear combination of elements in the fundamental sequence. Source points are placed outside of the solution interval in space, and by collocating at the endpoints of this interval a sequence of linear equations is obtained for finding the unknown coeficients. Tikhonov regularization is used to render a stable solution to the obtained systems of linear equations. Numerical results both for the parabolic and hyperbolic case confirm the efficiency of the proposed method including noisy data. The presented results complement the higher-dimensional case initiated in our previous researches.

Double homoclinic bifurcations by perturbing a class of cubic Z2-equivariant polynomial systems with nilpotent singular points

This paper focuses on the study of the limit cycle bifurcation of a general cubic Z2-equivariant Hamiltonian differential system with two nilpotent singular points. Initially, the necessary and sufficient conditions for the occurrence of two nilpotent singularities are provided in the unperturbed system, along with a complete description of all possible phase portraits on the plane. Subsequently, using a combination of the Melnikov function method and various analytical techniques, we derive the approximate expansion of the first order Melnikov function at the interval endpoint, as well as formula for its coefficients. These results can be utilized to analyze double homoclinic loop bifurcation of the perturbed system, finding the lower bound of limit cycles.

A Phase 1 Study to Assess the Effect of Supratherapeutic Doses of Iptacopan on Cardiac Conduction Parameters and Safety

Introduction: Iptacopan is a first-in-class, oral, selective inhibitor of factor B, a key component of the complement system alternative pathway. Inhibition of the complement system represents a potential mechanism for treatment of several diseases including paroxysmal nocturnal hemoglobinuria and atypical hemolytic uremic syndrome. The purpose of this randomized, participant-blinded, placebo-controlled phase 1 study was to assess the effect of supratherapeutic doses on the exposure-response relationship of iptacopan concentration and cardiac conduction, and to assess the cardiac safety and tolerability of ascending, supra-therapeutic single doses of iptacopan. Methods: Supratherapeutic doses of iptacopan 400, 800, and 1200 mg were included in this study, representing ≤6-fold higher dose than the planned clinical dose of 200 mg. The primary endpoints were change from baseline in Fridericia-corrected QT interval (QTcF) and safety endpoints. The analyses included data from this study pooled with part 1 data (single dose) from the first-in-human (CLNP023X2101) study (n=86). The study population included healthy men and women aged 18-55 years. The relationship between iptacopan plasma concentration and change from baseline in QTcF was estimated using a linear mixed effects model. Results: Thirty-two patients were randomized and received placebo (n=8) or iptacopan (n=24). The median age of participants in the 400-mg, 800-mg, 1200-mg, and placebo groups was 51.5, 35, 36, and 37 years, respectively. The upper bound of the 2-sided 90% CI for linear mixed effects model-derived placebo-adjusted change from baseline in QTcF at the geometric mean C max was 2.82 ms for 400-mg, 2.94 ms for 800-mg, and 3.22 ms for 1200-mg doses. Hence, there was no evidence of clinically meaningful QTc prolongation ( Table). The exposure-response analysis of PR and QRS intervals showed no iptacopan concentration-related effects. While the pre-specified and more simple exposure-response analysis of heart rate using change from day 1 baseline (0 hour) revealed an increase of 3 to 5 beats per minute, in a post hoc analysis using 24-hour, time-matched comparison for each participant at each time point before and after study treatment (the more common and gold-standard analysis), iptacopan had no effect on heart rate. No deaths, no moderate or severe treatment-emergent adverse events, and no treatment cohort discontinuations due to adverse events were reported. Conclusions: Three supratherapeutic doses were well tolerated and were not associated with any relevant change in QTcF. Iptacopan did not affect PR or QRS intervals and a time-matched post hoc analysis showed no effect on heart rate. These results indicate that the anticipated clinical dose of iptacopan (200 mg twice daily) has no adverse clinical effects on cardiac function.

Unit upper truncated Weibull distribution with extension to 0 and 1 inflated model – Theory and applications

A two-parameter unit distribution and its regression model plus its extension to 0 and 1 inflation is introduced and studied. The distribution is called the unit upper truncated Weibull (UUTW) distribution, while the inflated variant is called the 0−1 inflated unit upper truncated Weibull (ZOIUUTW) distribution. The UUTW distribution has an increasing and a J-shaped hazard rate function. The parameters of the proposed models are estimated by the method of maximum likelihood estimation. For the UUTW distribution, two practical examples involving household expenditure and maximum flood level data are used to show its flexibility and the proposed distribution demonstrates better fit tendencies than some of the competing unit distributions. Application of the proposed regression model demonstrates adequate capability in describing the real data set with better modeling proficiency than the existing competing models. Then, for the ZOIUUTW distribution, the CD34+ data involving cancer patients are analyzed to show the flexibility of the model in characterizing inflation at both endpoints of the unit interval.

Fiber denseness of intermediate β-shifts of finite type

We focus on ( 1), and . The -expansions of critical point are called kneading invariants, denoted as . Let with being periodic, we state that is a smooth curve which can be regarded as a fiber. By combinatorial method, we extend the results of Parry (1960 Acta Math. Acad. Sci. Hung. 11 401–16) and show that, the set of with its being a SFT is dense in . Similarly for the fiber . When considering another fiber , we demonstrate that when β is not a multinacci number, there are only countably many distinct matching intervals on . Using Markov approximation, we prove that the set of with being a SFT is dense in each matching interval. We also propose a classification scheme for the endpoints of these matching intervals.

Ranking of Fuzzy Numbers on the Basis of New Fuzzy Distance

Real-world problems often deal with uncertainties related to imprecision and vagueness, for which fuzzy modeling has provided a successful approach. Ranking fuzzy numbers is a necessary and complex task in multiple processes, such as decision making. To facilitate the task of ranking fuzzy numbers, it has been introduced an approach to construct fuzzy distances from classical interval distances because their α-cuts are continuous intervals. Usually, extant interval distances use interval endpoints or midpoints, leading to results that might not reflect the correct distance because of information loss. This paper introduces a new fuzzy distance based on a novel interval distance that considers all points within the intervals by using the concept of integration to calculate the average distance between all points belonging to two intervals, respectively. Subsequently, a series of distances between fuzzy numbers based on the proposed interval distances are defined and proved. A new method for ranking fuzzy numbers using the new fuzzy distances is then presented. Finally, the validity and effectiveness of the proposed distances will be demonstrated by a comparative analysis of numerical examples.

Enhanced trapezoidal rule for discontinuous functions

In many applications, data to be integrated is available only in the form of function values at predetermined equispaced points. For smooth periodic functions, the trapezoidal rule gives very accurate approximations to the integral, but is only second order accurate in more commonly arising non-periodic cases. The Gregory approach for end corrections can improve the accuracy order to 9 before the onset of negative weights, and of rapidly increasing ill-conditioning. In recent work, this has been extended to accuracy orders around 20.This present study focuses on the more general case when one or both end points of the integration interval do not coincide with any of the equispaced grid points. We show that accuracy orders up to around 10 can also then be achieved, with all quadrature weights still remaining non-negative. This method can be utilized for example when functions to be integrated feature discontinuities at locations that can be separately determined (at or in between the equispaced grid points).