Usual Regularity Research Articles

Generalized versus classical normal derivative

"Given a bounded domain with Lipschitz boundary, the general Green formula permits to justify that the weak solutions of a Neumann elliptic problem satisfy the Neumann boundary condition in a weak sense. The formula involves a generalized normal derivative. We prove a general result which establishes that the generalized normal derivative of an operator coincides with the classical one, provided that the operator is continuous. This result allows to deduce that, under usual regularity assumptions, the weak solutions of a Neumann problem satisfy the Neumann boundary condition in the classical sense. This information is nec- essary in particular for applying the strong maximum principle."

Menstrual cycle characteristics and incident cancer: a prospective cohort study.

Are menstrual cycle characteristics throughout the reproductive lifespan associated with cancer risk? Irregular and long menstrual cycles throughout the reproductive lifespan were associated with increased risk of total invasive cancer, especially obesity-related cancers. Long and irregular menstrual cycles have been associated with lower risk of pre-menopausal breast cancer and higher risk of endometrial cancer, but associations with other malignancies are less clear. Prospective cohort study. Prospective follow-up of 78 943 women participating in the Nurses' Health Study II between 1989 and 2015. We followed 78 943 pre-menopausal women without cancer history who reported the usual length and regularity of their menstrual cycles at different ages (14-17, 18-22 and 29-46 years). Cancer diagnosis was confirmed through medical record review and classified as obesity-related (colorectal, gallbladder, kidney, multiple myeloma, thyroid, pancreatic, esophageal, gastric, liver, endometrial, ovarian and post-menopausal breast) or non-obesity-related. We fitted Cox proportional hazards models to estimate hazard ratios (HRs) and 95% CIs of the association between menstrual cycle characteristics and cancer incidence. We documented 5794 incident cancer cases during 1 646 789 person-years of follow-up. After adjusting for BMI and other potential confounders, women reporting irregular cycles at age 29-46 years had an 11% (95% CI: 2-21%) higher risk of total invasive cancer than women reporting very regular cycles at the same age. This association was limited to obesity-related cancers, with a 23% (95% CI: 9-39%) higher risk and was strongest for endometrial cancer (HR = 1.39; 95% CI: 1.09-1.77). Findings were comparable for cycle characteristics earlier in life and for menstrual cycle length. Very irregular cycles at age 14-17 years were associated with significant increase in risk of colorectal cancer (HR = 1.36; 95% CI: 1.02-1.81). Our study might be subject to recall bias for findings pertaining to cycle characteristics in adolescence and early adulthood, as these were retrospectively reported. Generalizability to non-White women may be limited, as 96% of participants were White. Women with irregular or long menstrual cycles in mid-adulthood had a statistically significantly higher risk of developing cancer, especially obesity-related cancers. This association was not limited to gynecological cancers. Obesity-related cancers may need to be added to the spectrum of long-term health consequences of long or irregular cycles, possibly warranting targeted screening among women who experience long or irregular cycles in mid-adulthood. This work was supported by grants U01 CA176726, U01 HL145386 and R01 HD096033 from the National Institutes of Health. The authors have no conflicts of interest to declare. N/A.

Proportional likelihood ratio mixed model for discrete longitudinal data.

Rathouz and Gao [2] and Luo and Tsai [3] proposed valuable extensions to the generalized linear model for modeling a nonlinear monotonic relationship between the mean response and a set of covariates. In their extensions for discrete data the baseline response distribution is unspecified and is estimated from the data. We propose to extend this model for the analysis of longitudinal data by incorporating random effects into the linear predictor, and using maximum likelihood for estimation and inference. Motivated in particular by longitudinal studies of clinical scale outcomes, we developed an estimation procedure for a finite-support response using a generalized expectation-maximization algorithm where Gauss-Hermite quadrature is employed to approximate the integrals in the E step of the algorithm. Upon convergence, the observed information matrix is estimated through second-order numerical differentiation of the log-likelihood function. Asymptotic properties of the maximum likelihood estimates follow under the usual regularity conditions. Simulation studies are conducted to assess its finite-sample properties and compare the proposed model to the generalized linear mixed model. The proposed method is illustrated in an analysis of data from a longitudinal study of Huntington's disease.

Error analysis of symmetric linear/bilinear partially penalized immersed finite element methods for Helmholtz interface problems

This article presents an error analysis of the symmetric linear/bilinear partially penalized immersed finite element (PPIFE) methods for interface problems of Helmholtz equations. Under the assumption that the exact solution possesses a usual piecewise H2 regularity, the optimal error bounds for the PPIFE solutions are derived in an energy norm and the usual L2 norm. A numerical example is conducted to validate the theoretical conclusions.

Associations of Menstrual Cycle Characteristics Across the Reproductive Life Span and Lifestyle Factors With Risk of Type 2 Diabetes

Menstrual cycle dysfunction is associated with insulin resistance, a key feature early in the pathogenesis of type 2 diabetes. However, the evidence linking irregular and long menstrual cycles with type 2 diabetes is scarce and inconsistent. To evaluate the associations between menstrual cycle characteristics at different points throughout a woman's reproductive life span and risk of type 2 diabetes and the extent to which this association is modified by lifestyle factors. This prospective cohort study included 75 546 premenopausal US female nurses participating in the Nurses' Health Study II from 1993 to June 30, 2017. Data analysis was performed from February 1 to December 30, 2019. Self-reported usual length and regularity of menstrual cycles at the age ranges of 14 to 17 years, 18 to 22 years, and 29 to 46 years. Incident type 2 diabetes identified through self-report and confirmed by validated supplemental questionnaires. Among the 75 546 women in the study at baseline, the mean (SD) age was 37.9 (4.6) years (range, 29.0-46.0 years). A total of 5608 participants (7.4%) had documented new cases of type 2 diabetes during 1 639 485 person-years of follow-up. After adjustment for potential confounders, women reporting always having irregular menstrual cycles between the age ranges of 14 to 17 years, 18 to 22 years, and 29 to 46 years were, respectively, 32% (95% CI, 22%-44%), 41% (95% CI, 23%-62%), and 66% (95% CI, 49%-84%) more likely to develop type 2 diabetes than women reporting very regular cycles (within 3-4 days of expected period) in the same age range. Similarly, women reporting a usual cycle length of 40 days or more between the age ranges of 18 to 22 years and 29 to 46 years were, respectively, 37% (95% CI, 19%-57%) and 50% (95% CI, 36%-65%) more likely to develop type 2 diabetes during follow-up compared with women reporting a usual cycle length of 26 to 31 days in the same age ranges. These associations appeared to be stronger among women with overweight or obesity, a low-quality diet, and low levels of physical activity. The relative excess risk of type 2 diabetes due to the interaction between irregular and long menstrual cycles and the overall unhealthy lifestyle score was 0.73 (95% CI, 0.57-0.89) and 0.68 (95% CI, 0.54-0.83), respectively. In this cohort study of US female nurses participating in the Nurses' Health Study II, irregular and long menstrual cycles throughout life were associated with a greater risk of type 2 diabetes, particularly among women with overweight or obesity, a low-quality diet, and low levels of physical activity.

Menstrual cycle regularity and length across the reproductive lifespan and risk of premature mortality: prospective cohort study.

ObjectiveTo evaluate whether irregular or long menstrual cycles throughout the life course are associated with all cause and cause specific premature mortality (age <70 years).DesignProspective cohort study.SettingNurses’ Health Study II (1993-2017).Participants79 505 premenopausal women without a history of cardiovascular disease, cancer, or diabetes and who reported the usual length and regularity of their menstrual cycles at ages 14-17 years, 18-22 years, and 29-46 years.Main outcome measuresHazard ratios and 95% confidence intervals for all cause and cause specific premature mortality (death before age 70 years) were estimated from multivariable Cox proportional hazards models.ResultsDuring 24 years of follow-up, 1975 premature deaths were documented, including 894 from cancer and 172 from cardiovascular disease. Women who reported always having irregular menstrual cycles experienced higher mortality rates during follow-up than women who reported very regular cycles in the same age ranges. The crude mortality rate per 1000 person years of follow-up for women reporting very regular cycles and women reporting always irregular cycles were 1.05 and 1.23 for cycle characteristics at ages 14-17 years, 1.00 and 1.37 for cycle characteristics at ages 18-22 years, and 1.00 and 1.68 for cycle characteristics at ages 29-46 years. The corresponding multivariable adjusted hazard ratios for premature death during follow-up were 1.18 (95% confidence interval 1.02 to 1.37), 1.37 (1.09 to 1.73), and 1.39 (1.14 to 1.70), respectively. Similarly, women who reported that their usual cycle length was 40 days or more at ages 18-22 years and 29-46 years were more likely to die prematurely than women who reported a usual cycle length of 26-31 days in the same age ranges (1.34, 1.06 to 1.69; and 1.40, 1.17 to 1.68, respectively). These relations were strongest for deaths related to cardiovascular disease. The higher mortality associated with long and irregular menstrual cycles was slightly stronger among current smokers.ConclusionsIrregular and long menstrual cycles in adolescence and adulthood are associated with a greater risk of premature mortality (age <70 years). This relation is slightly stronger among women who smoke.

Solving nonlinear non-local problems using positive square-root operators

A non-constructive existence theory for certain operator equations L u = D u , using the substitution u = B 1 2 ξ with B = L −1 , is developed, where L is a linear operator (in a suitable Banach space) and D is a homogeneous nonlinear operator such that Dλu = λ α D u for all λ ≥ 0 and some α ∈ R , α ≠ ~1. This theory is based on the positive-operator approach of Krasnosel’skii. The method has the advantage of being able to tackle the nonlinear right-hand side D in cases where conventional operator techniques fail. By placing the requirement that the operator B must have a positive square root, it is possible to avoid the usual regularity condition on either the mapping D or its Fréchet derivative. The technique can be applied in the case of elliptic PDE problems, and we show the existence of solitary waves for a generalization of Benjamin’s fluid dynamics problem.

Equilibrium Non-Existence in Generalized Games

A generalized game is a strategic situation in which agents’ behavior restricts their opponents’ available action choices, giving rise to interdependencies in terms of what strategy profiles remain mutually feasible. This paper proposes a novel example of a simple generalized game in which the well-known convexity, compactness, continuity, and concavity assumptions are satisfied, but no Nash equilibrium exists. This finding reinforces that certain additional conditions appearing in the literature (primarily, the lower hemicontinuity of feasibility correspondences) are indispensable for equilibrium existence and must be considered as supplemental desiderata beyond the usual regularity conditions. Implications for institutional design are discussed.

Menstrual Cycle Characteristics Across the Reproductive Lifespan, Lifestyle Factors and Risk of Type 2 Diabetes

Background: Menstrual cycle dysfunction is associated with insulin resistance; however, the evidence linking irregular/long menstrual cycles with type 2 diabetes (T2D) is scarce and inconsistent. It is also unclear whether this relation can be modified by lifestyle factors. Methods: Between 1993 and 2017, we followed 75,546 premenopausal women without T2D at baseline who reported the usual length and regularity of their menstrual cycles at ages of 14-17, 18-22, and 29-46 years, as well as data on body weight, smoking status, physical activity, and diet every 2-4 years during follow-up. Time-dependent Cox proportional hazards regression models were constructed to estimate the hazard ratios (HRs) and 95% confidential intervals (CI) for the associations between menstrual cycle characteristics and T2D risk. Findings: We documented 5,608 new cases of T2D. After adjusting for potential confounders including BMI and lifestyle factors, women reporting always having irregular menstrual cycles between the ages of 14-17, 18-22, and 29-46 years were, respectively, 33% (95% CI: 23, 45%), 42% (95% CI: 23, 63%), and 66% (95% CI: 49, 84%) more likely to develop T2D than women reporting very regular cycles in the same age range. Similarly, women reporting usual cycle length ≥40 days between the ages of 18-22 and 29-46 years also had a higher risk of T2D than women reporting a usual cycle length of 26-31 days in the same age range (HRs=1.37; 95% CI: 1.20, 1.58 and 1.49; 95% CI: 1.36, 1.64, respectively). These relations were stronger for women with overweight/obesity and a lower level of physical activity when additive interaction was evaluated. Interpretation : Irregular and long menstrual cycles throughout life are associated with a higher risk of T2D, with a particularly elevated risk among women who were overweight/obesity and physical inactivity. Funding Statement: The National Institutes of Health grants U01-HL145386, U01-CA176726, R01-HL034594, R01-HL088521, DK120870, and DK112940. Declaration of Interests: The authors declare no competing interest. Ethics Approval Statement: Study procedures have been approved by the institutional review boards of the Brigham and Women’s Hospital and Harvard T.H. Chan School of Public Health.

The multiscale hybrid mixed method in general polygonal meshes

This work extends the general form of the multiscale hybrid-mixed (MHM) method for the second-order Laplace (Darcy) equation to general non-conforming polygonal meshes. The main properties of the MHM method, i.e., stability, optimal convergence, and local conservation, are proven independently of the geometry of the elements used for the first level mesh. More precisely, it is proven that piecewise polynomials of degree k and $$k+1$$, $$k \ge 0$$, for the Lagrange multipliers (flux), along with continuous piecewise polynomial interpolations of degree $$k+1$$ posed on second-level sub-meshes are stable if the latter is fine enough with respect to the mesh for the Lagrange multiplier. We provide an explicit sufficient condition for this restriction. Also, we prove that the error converges with order $$k+1$$ and $$k+2$$ in the broken $$H^1$$ and $$L^2$$ norms, respectively, under usual regularity assumptions, and that such estimates also hold for non-convex; or even non-simply connected elements. Numerical results confirm the theoretical findings and illustrate the gain that the use of multiscale functions provides.

Automorphy of $m$-fold tensor products of GL(2)

We prove that for any m &gt; 1 , given any m -tuple of Hecke eigenforms f_i of level 1 whose weights satisfy the usual regularity condition, there is a self-dual cuspidal automorphic form \pi of {\rm GL}_{2^m}(\mathbb{Q}) corresponding to their tensor product, i.e., such that the system of Galois representations attached to \pi agrees with the tensor product of the ones attached to the cuspforms f_i .

Density of positive Lyapunov exponents for symplectic cocycles

We prove that Sp (2d,\mathbb R) -cocycles, HSp( 2d )-cocycles and pseudo-unitary cocycles with at least one non-zero Lyapunov exponent are dense in all usual regularity classes for non periodic dynamical systems. For Schrödinger operator on the strip, we prove a similar result for density of positive Lyapunov exponents. This generalizes a result of A. Avila [2] to higher dimensions.

Analysis and numerical solution of a nonlinear variable-order fractional differential equation

We prove the wellposedness of a nonlinear variable-order fractional differential equation and the regularity of its solutions. The regularity of the solutions is determined solely by the values of the variable order and its high-order derivatives at time t = 0 (in addition to the usual regularity assumptions on the variable order and the coefficients). If the variable-order reduces to an integer order at t = 0, then the solution has full regularity as the solution to a first-order ordinary differential equation. In this case, we prove that the corresponding finite difference scheme discretized on a uniform mesh has an optimal-order convergence rate. However, if the variable order does not reduce to an integer order at t = 0, then the solution has a singularity at time t = 0, as Stynes et al. proved in [15] for the constant-order time-fractional diffusion equations. The corresponding finite difference scheme discretized on a uniform mesh has only a suboptimal-order convergence rate. Instead, we prove that the finite difference scheme discretized on a graded mesh determined by the value of the variable order at time t = 0 has an optimal-order convergence rate in terms of the number of the time steps. Numerical experiments substantiate these theoretical results.

Uniform Shapiro-Lopatinski Conditions and Boundary Value Problems on Manifolds with Bounded Geometry

We study the regularity of the solutions of second order boundary value problems on manifolds with boundary and bounded geometry. We first show that the regularity property of a given boundary value problem $(P, C)$ is equivalent to the uniform regularity of the natural family $(P_x, C_x)$ of associated boundary value problems in local coordinates. We verify that this property is satisfied for the Dirichlet boundary conditions and strongly elliptic operators via a compactness argument. We then introduce a uniform Shapiro-Lopatinski regularity condition, which is a modification of the classical one, and we prove that it characterizes the boundary value problems that satisfy the usual regularity property. We also show that the natural Robin boundary conditions always satisfy the uniform Shapiro-Lopatinski regularity condition, provided that our operator satisfies the strong Legendre condition. This is achieved by proving that "well-posedness implies regularity" via a modification of the classical "Nirenberg trick". When combining our regularity results with the Poincare inequality of (Ammann-Grosse-Nistor, preprint 2015), one obtains the usual well-posedness results for the classical boundary value problems in the usual scale of Sobolev spaces, thus extending these important, well-known theorems from smooth, bounded domains, to manifolds with boundary and bounded geometry. As we show in several examples, these results do not hold true anymore if one drops the bounded geometry assumption. We also introduce a uniform Agmon condition and show that it is equivalent to the coerciveness. Consequently, we prove a well-posedness result for parabolic equations whose elliptic generator satisfies the uniform Agmon condition.

Constrained dynamics of tachyon field in FRWL spacetime

In this paper, we continue the study of tachyon scalar field described by a Dirac-Born-Infeld (DBI) type action with constraints in the cosmological context. The proposed extension of the system introducing an auxiliary field in the minisuperspace framework is discussed. A new equivalent set of constraints is constructed, satisfying the usual regularity conditions.

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The Uncovered Set and the Core: Cox's Result Revisited

In this work first it is shown, in contradiction to the well-known claim in Cox (1987), that the uncovered set in a multidimensional spatial voting situation (under the usual regularity conditions) does not necessarily coincide with the core even when the core is singleton: in particular, the posited coincidence result, while true for an odd number of voters, may cease to be true when the number of voters is even. Second we provide a characterisation result for the case with an even number of voters: a singleton core is the uncovered set in this case if and only if the unique element in the core is the Condorcet winner.

A Hopf–Lax formula for the time evolution of the level-set equation and a new approach to shape sensitivity analysis

The level-set method is used in many different applications to describe the propagation of shapes and domains. When scalar speed fields are used to encode the desired shape evolution, this leads to the classical level-set equation. We present a concise Hopf–Lax representation formula that can be used to characterise the evolved domains at arbitrary times. This result is also applicable for the case of speed fields without a fixed sign, even though the level-set equation has a non-convex Hamiltonian in these situations. The representation formula is based on the same idea that underpins the Fast-Marching Method, and it provides a strong theoretical justification for a generalised Composite Fast-Marching method. Based on our Hopf–Lax formula, we are also able to present new theoretical results. In particular, we show non-fattening of the zero level set in a measure-theoretic sense, derive a very general shapesensitivity calculus that does not require the usual regularity assumptions on the domains, prove optimal Lipschitz constants for the evolved level-set function and discuss the effect of perturbations in both the speed field and the initial geometry.

Invariant tests based on M-estimators, estimating functions, and the generalized method of moments

ABSTRACTWe study the invariance properties of various test criteria which have been proposed for hypothesis testing in the context of incompletely specified models, such as models which are formulated in terms of estimating functions (Godambe, 1960) or moment conditions and are estimated by generalized method of moments (GMM) procedures (Hansen, 1982), and models estimated by pseudo-likelihood (Gouriéroux, Monfort, and Trognon, 1984b,c) and M-estimation methods. The invariance properties considered include invariance to (possibly nonlinear) hypothesis reformulations and reparameterizations. The test statistics examined include Wald-type, LR-type, LM-type, score-type, and C(α)−type criteria. Extending the approach used in Dagenais and Dufour (1991), we show first that all these test statistics except the Wald-type ones are invariant to equivalent hypothesis reformulations (under usual regularity conditions), but all five of them are not generally invariant to model reparameterizations, including measurement unit changes in nonlinear models. In other words, testing two equivalent hypotheses in the context of equivalent models may lead to completely different inferences. For example, this may occur after an apparently innocuous rescaling of some model variables. Then, in view of avoiding such undesirable properties, we study restrictions that can be imposed on the objective functions used for pseudo-likelihood (or M-estimation) as well as the structure of the test criteria used with estimating functions and generalized method of moments (GMM) procedures to obtain invariant tests. In particular, we show that using linear exponential pseudo-likelihood functions allows one to obtain invariant score-type and C(α)−type test criteria, while in the context of estimating function (or GMM) procedures it is possible to modify a LR-type statistic proposed by Newey and West (1987) to obtain a test statistic that is invariant to general reparameterizations. The invariance associated with linear exponential pseudo-likelihood functions is interpreted as a strong argument for using such pseudo-likelihood functions in empirical work.

Parametric Raviart--Thomas Elements for Mixed Methods on Domains with Curved Surfaces

The finite element approximation on curved boundaries using parametric Raviart--Thomas spaces is studied in the context of the mixed formulation of Poisson's equation as a saddle-point system. It is shown that optimal-order convergence is retained on domains with piecewise $C^{k+2}$ boundary for the parametric Raviart--Thomas space of degree $k \geq 0$ under the usual regularity assumptions. This extends the analysis in [F. Bertrand, S. Munzenmaier, and G. Starke, SIAM J. Numer. Anal., 52 (2014), pp. 3165--3180] from the first-order system least squares formulation to mixed approaches of saddle-point type. In addition, a detailed proof of the crucial estimate in three dimensions is given which handles some complications not present in the two-dimensional case. Moreover, the appropriate treatment of inhomogeneous flux boundary conditions is discussed. The results are confirmed by computational results which also demonstrate that optimal-order convergence is not achieved, in general, if standard Raviart--Th...