Weak Regularity Research Articles

784 Naturalistic Measurement of Sleep/Wake Disturbance in Adults Receiving Methadone Treatment for Opioid Use Disorder

Abstract Introduction Individuals with opioid use disorder (OUD) report significant sleep/wake disturbance, which continues even after stabilization on medication-assisted treatment (MAT). However, the nature of sleep/wake disturbance in this population has not been well documented objectively. Here we analyze naturalistic wrist actigraphy recordings in individuals with OUD receiving methadone-based MAT. Methods Seven adults undergoing methadone treatment for OUD (ages 26–50; 4 women) wore a wrist actigraph (Actiwatch-2, Philips Respironics) continuously for 7 days. They were asked to adhere to their normal sleep schedule in order to obtain naturalistic observations. Reference data were collected in a separate study of healthy controls, in which 14 hospital nurses (ages 20–60; 13 women) wore a wrist actigraph continuously for 7 days. In this reference group, 7 participants had a day shift schedule (07:00–19:00) and 7 had a night shift schedule (19:00–07:00), with six 12h shifts in a 2-week period. Actigraphic data were collected in 1min epochs, and the sleep/wake status for each epoch was estimated using Actiware 6.0.9 (Philips Respironics). The estimated sleep/wake patterns were subjected to cosinor analysis to assess 24h rhythmicity and analysis of the distribution of inactive periods to assess sleep continuity. Results For the reference group, nurses working day shifts displayed strong 24h rhythmicity, whereas nurses working night shifts showed blunted 24h rhythmicity (F[1,12]=66.11, p<0.001). However, both day and night nurses exhibited high sleep continuity (KS test, p=0.82). By contrast, for the group with OUD receiving methadone, the strength of 24h rhythmicity was reduced to between that of the day and night shift nurses in the reference group, indicating weak regularity of sleep/wake patterns (F[2,18]=33.79, p<0.001). Furthermore, individuals with OUD receiving methadone experienced low sleep continuity compared to the reference group (KS test, p=0.030). Conclusion These naturalistic observations confirm the presence of sleep/wake disturbance, resulting from both irregular sleep/wake patterns and low sleep continuity, in individuals receiving methadone-based MAT for OUD. Sleep/wake disturbance may interfere with the ability to achieve OUD recovery goals, and comparing sleep disturbance in MAT populations to reference data highlights the need to consider sleep in these populations as a clinical priority. Support (if any) State of Washington Initiative Measure No. 171

Efficient difference method for time-space fractional diffusion equation with Robin fractional derivative boundary condition

In this paper, a numerical method is proposed to solve the time-space fractional diffusion equation with Robin fractional derivative boundary condition. Under the weak regularity assumptions of solution, we present a numerical scheme based on the L1 method for time discretization on graded mesh and the Grunwald-Letnikov formula for spatial discretization on uniform mesh. And a fast scheme for the considered problem is constructed based on the exponential summation approximation of the kernel function t−α. Meanwhile, a detailed analysis of stability and convergence is given. Then, the extrapolation method is applied to the space direction to make it reach the second-order accuracy. Finally, numerical experiments show that the proposed method is effective.

Prediction-based estimation for diffusion models with high-frequency data

This paper obtains asymptotic results for parametric inference using prediction-based estimating functions when the data are high-frequency observations of a diffusion process with an infinite time horizon. Specifically, the data are observations of a diffusion process at n equidistant time points \(\Delta _n i\), and the asymptotic scenario is \(\Delta _n \rightarrow 0\) and \(n\Delta _n \rightarrow \infty\). For useful and tractable classes of prediction-based estimating functions, existence of a consistent estimator is proved under standard weak regularity conditions on the diffusion process and the estimating function. Asymptotic normality of the estimator is established under the additional rate condition \(n\Delta _n^3 \rightarrow 0\). The prediction-based estimating functions are approximate martingale estimating functions to a smaller order than what has previously been studied, and new non-standard asymptotic theory is needed. A Monte Carlo method for calculating the asymptotic variance of the estimators is proposed.

Temporal second-order difference methods for solving multi-term time fractional mixed diffusion and wave equations

This article deals with an establishment and sharp theoretical analysis of a numerical scheme devised for solving the multi-dimensional multi-term time fractional mixed diffusion and wave equations. The governing equation contains both fractional diffusion term and fractional wave term which make the numerical analysis challenging. With the help of the method of order reduction, we convert the time multi-term fractional diffusion and wave terms into the time multi-term fractional integral and diffusion terms respectively, and then develop L2-1σ formula for solving the latter problem. In addition, the formula is used to numerically solve the time distributed-order diffusion and wave equations. The stability and convergence of these numerical schemes are rigorously analyzed by the energy method. The convergence rates are of order two in both time and space. A difference scheme on nonuniform time grids is also constructed for solving the problem with weak regularity at the initial time. Finally, we illustrate our results with some examples.

A polygonal discontinuous Galerkin method with minus one stabilization

We propose a discontinuous Galerkin method for the Poisson equation on polygonal tessellations in two dimensions, stabilized by penalizing, locally in each elementK, a residual term involving the fluxes, measured in the norm of the dual ofH1(K). The scalar product corresponding to such a norm is numerically realizedviathe introduction of a (minimal) auxiliary space inspired by the Virtual Element Method. Stability and optimal error estimates in the brokenH1norm are proven under a weak shape regularity assumption allowing the presence of very small edges. The results of numerical tests confirm the theoretical estimates.

On mathematical analysis of complex fluids in active hydrodynamics

<p style='text-indent:20px;'>This is a survey article for this special issue providing a review of the recent results in the mathematical analysis of active hydrodynamics. Both the incompressible and compressible models are discussed for the active liquid crystals in the Landau-de Gennes Q-tensor framework. The mathematical results on the weak solutions, regularity, and weak-strong uniqueness are presented for the incompressible flows. The global existence of weak solution to the compressible flows is recalled. Other related results on the inhomogeneous flows, incompressible limits, and stochastic analysis are also reviewed.</p>

A prediction method for the service trajectory of badminton moving video based on fuzzy clustering algorithm

Due to the weak regularity of target's motion attitude in badminton video, the existing methods cannot segment the image accurately, which leads to the low accuracy of predicting the service track of badminton video. A method of predicting the service track of badminton based on fuzzy clustering algorithm is proposed. The fuzzy clustering algorithm is used to segment the video image of badminton movement, obtain the contour of the serving arm in the video image, calculate the body state ratio and compactness of the serving arm of the moving object, construct the overall matching similarity function of the moving object, and realise the tracking and trajectory prediction of the moving object in the video image. The experimental results show that the proposed method has good noise iteration performance, clearer target of the processed motion video image, and can accurately predict the serve trajectory in the badminton motion video.

Spectral Analysis of One-Dimensional Dirac System with Summable Potential and Sturm- Liouville Operator with Distribution Coefficients

We consider one-dimensional Dirac operatorLP,U with Birkhoff regular boundary conditions and summable potential P(x) on[0, ]. We introduce strongly and weakly regular operators. In both cases, asymptotic formulas for eigenvalues are found. In these formulas, we obtain main asymptotic terms and estimates for the second term. We specify these estimates depending on the functional class of the potential: Lp[0,] with p [1,2] and the Besov space Bp,p'[0,] with p [1,2] and (0,1/p). Additionally, we prove that our estimates are uniform on balls Pp,R Then we get asymptotic formulas for normalized eigenfunctions in the strongly regular case with the same residue estimates in uniform metric on x [0,]. In the weakly regular case, the eigenvalues 2n and 2n+1 are asymptotically close and we obtain similar estimates for two-dimensional Riesz projectors. Next, we prove the Riesz basis property in the space (L2[0,])2 for a system of eigenfunctions and associated functions of an arbitrary strongly regular operatorLP,U. In case of weak regularity, the Riesz basicity of two-dimensional subspaces is proved.
 In parallel with theLP,U operator, we consider the SturmLiouville operator Lq,U generated by the differential -y'' + q(x)y expressionwith distribution potential q of first-order singularity (i.e., we assume that the primitive u = q(1) belongs to L2[0, ]) and Birkhoff-regular boundary conditions. We reduce to this case -(1y')'+i(y)'+iy'+0y, operators of more general form where '1,,0(-1)L2and 10. For operator Lq,U, we get the same results on the asymptotics of eigenvalues, eigenfunctions, and basicity as for operator LP,U .
 Then, for the Dirac operator LP,U, we prove that the Riesz basis constant is uniform over the ballsPp,R for p1 or 0. The problem of conditional basicity is naturally generalized to the problem
 of equiconvergence of spectral decompositions in various metrics. We prove the result on equiconvergence by varying three indices: fL[0,] (decomposable function), PL[0,] (potential), and Sm-Sm00,m in L[0,] (equiconvergence of spectral decompositions in the corresponding norm). In conclusion, we prove theorems on conditional and unconditional basicity of the system of eigenfunctions and associated functions of operator LP,U in the spaces L[0,],2, and in various Besov spaces Bp,q[0,].

Mathematical Analysis and the Local Discontinuous Galerkin Method for Caputo–Hadamard Fractional Partial Differential Equation

In this paper, we study the Caputo–Hadamard fractional partial differential equation where the time derivative is the Caputo–Hadamard fractional derivative and the space derivative is the integer-order one. We first introduce a modified Laplace transform. Then using the newly defined Laplace transform and the well-known finite Fourier sine transform, we obtain the analytical solution to this kind of linear equation. Furthermore, we study the regularity and logarithmic decay of its solution. Since the equation has a time fractional derivative, its solution behaves a certain weak regularity at the initial time. We use the finite difference scheme on non-uniform meshes to approximate the time fractional derivative in order to guarantee the accuracy and use the local discontinuous Galerkin method (LDG) to approximate the spacial derivative. The fully discrete scheme is established and analyzed. A numerical example is displayed which support the theoretical analysis.

Research on determining of cations in GeAsSeI chalcohalide glass

In order to improve the thermal stability of chalcohalide glasses, which is beneficial for fiber drawing, and blue-shifting zero-dispersion-wavelength (ZDW), which is beneficial to generate supercontinuum (SC) spectra, so more germanium (Ge) is needed to be incorporated into the glass network of Ge-As-Se-I. Here, two series of (Ge20AsxSe80-x)85I15 and (GexAs40-xSe60)85I15 glasses are investigated on the optical properties, including absorption edge at short wavelength, optical band gap, linear refractive index and ZDW. The glass transition temperature (Tg) and the soften temperature (Tp) of these glasses are also investigated. In general, as the ratio of Ge/As changes, optical properties of (GexAs40-xSe60)85I15 glasses change regularly, but when the ratio of As/Se changes, those of the (Ge20AsxSe80-x)85I15 glasses keep weak regularity relatively. Generally, the effect of Ge on Ge-As-Se-I glasses’ properties is well presented in this work, and this will show much help to develop novel mid-infrared fiber and device in the future.

Carbon, oxygen and strontium isotopes of the Mesoproterozoic Jixian System （1.6-1.4 Ga）in the southern margin of the North China Craton and the geological implications

ABSTRACT The Mesoproterozoic Jixian System (1.6–1.4 Ga) in the Luonan area, southern margin of the North China Craton (NCC), is characterized by complete development, continuous sedimentation and a low degree of structural deformation, resulting in well-preserved palaeontological and paleoenvironmental records. Carbon, oxygen and strontium isotopes of 382 dolostone samples from the Jixian System in Xiong’er Rift, southern margin of the NCC are analysed for the restoration of paleoenvironment and stratigraphic correlation. The results show that the δ13C values of these samples mainly concentrate between −1.8‰ and 2.0‰, and the values can be divided into three segments. The δ18O values are mainly between −6.5‰ and −3.5‰, with frequent fluctuations and weak regularity. The 87Sr/86Sr ratio ranges from 0.7051 to 0.7139, with an average value of 0.7085. Its fluctuations have certain regularity and are negatively correlated with the carbon isotopes, indicating that the sea-level changes during the Jixian period in the southern margin of the NCC. The carbon isotope curve of the samples from the study section is very similar to those of the Xiong’er Rift and the Yanliao Rift of the NCC. The −3.2‰ δ13C negative excursion in the middle and upper parts of the Xunjiansi Formation corresponds to the δ13C negative excursion from the top of the Wumishan Formation to the bottom of the Tieling Formation in the Yanliao Rift. Combined with previous research data, the Jixian System that deposited in the Xiong’er Rift and the Yanliao Rift can be approximately correlated with each other. In addition, the carbon isotope data obtained from this study is comparable to global carbon isotope values during the same period. The rise and oscillation of carbon and strontium isotope values during this period may be a response to the break-up of the Colombian supercontinent.

Finite-sample generalized confidence distributions and sign-based robust estimators in median regressions with heterogeneous dependent errors

We study the problem of estimating the parameters of a linear median regression without any assumption on the shape of the error distribution – including no condition on the existence of moments – allowing for heterogeneity (or heteroskedasticity) of unknown form, noncontinuous distributions, and very general serial dependence (linear and nonlinear). This is done through a reverse inference approach, based on a distribution-free sign-based testing theory, from which confidence sets and point estimators are subsequently generated. We propose point estimators, which have a natural association with confidence distributions. These estimators are based on maximizing test p-values and inherit robustness properties from the generating distribution-free tests. Both finite-sample and large-sample properties of the proposed estimators are established under weak regularity conditions. We show that they are median-unbiased (under symmetry and estimator unicity) and possess equivariance properties. Consistency and asymptotic normality are established without any moment existence assumption on the errors. A Monte Carlo study of bias and RMSE shows sign-based estimators perform better than LAD-type estimators in various heteroskedastic settings. We illustrate the use of sign-based estimators on an example of β-convergence of output levels across U.S. states.

On the Zeros of Non-Analytic Random Periodic Signals

Abstract In this paper, we investigate the local universality of the number of zeros of a random periodic signal of the form $S_n(t)=\sum _{k=1}^n a_k f(k t)$, where $f$ is a $2\pi -$periodic function satisfying weak regularity conditions and where the coefficients $a_k$ are i.i.d. random variables, which are centered with unit variance. In particular, our results hold for continuous piecewise linear functions. We prove that the number of zeros of $S_n(t)$ in a shrinking interval of size $1/n$ converges in law as $n$ goes to infinity to the number of zeros of a Gaussian process whose explicit covariance only depends on the function $f$ and not on the common law of the random coefficients $(a_k)$. As a byproduct, this entails that the point measure of the zeros of $S_n(t)$ converges in law to an explicit limit on the space of locally finite point measures on $\mathbb R$ endowed with the vague topology. The standard tools involving the regularity or even the analyticity of $f$ to establish such kind of universality results are here replaced by some high-dimensional Berry–Esseen bounds recently obtained in [ 7]. The latter allow us to prove functional Central Limit Theorems in $C^1$ or Lipschitz topology in situations where usual criteria cannot be applied due to the lack of regularity.

Szemerédi’s proof of Szemerédi’s theorem

In 1975, Szemeredi famously established that any set of integers of positive upper density contained arbitrarily long arithmetic progressions. The proof was extremely intricate but elementary, with the main tools needed being the van der Waerden theorem and a lemma now known as the Szemeredi regularity lemma, together with a delicate analysis (based ultimately on double counting arguments) of limiting densities of sets along multidimensional arithmetic progressions. In this note we present an arrangement of this proof that incorporates a number of notational and technical simplifications. Firstly, we replace the use of the regularity lemma by that of the simpler ``weak regularity lemma'' of Frieze and Kannan. Secondly, we extract the key inductive steps at the core of Szemeredi's proof (referred to as ``Lemma 5'', ``Lemma 6'', and ``Fact 12'' in that paper) as stand-alone theorems that can be stated with less notational setup than in the original proof, in particular involving only (families of) one-dimensional arithmetic progressions, as opposed to multidimensional arithmetic progressions. Thirdly, we abstract the analysis of limiting densities along the (now one-dimensional) arithmetic progressions by introducing the notion of a family of arithmetic progressions with the ``double counting property''. We also present a simplified version of the argument that is capable of establishing Roth's theorem on arithmetic progressions of length three.

Convergence analysis of Crank-Nicolson extrapolated fully discrete scheme for thermally coupled incompressible magnetohydrodynamic system

In this paper, we consider the nonstationary magnetohydrodynamic coupled heat equation through the well-known Boussinesq approximation. The Crank-Nicolson extrapolation scheme is used for time derivative terms, and the mixed finite method is used for spatial discretization. We employ the Taylor-Hood finite elements to approximate Navier-Stokes equations, Nédélec edge element for the magnetic induction and the equal order Lagrange elements for the thermal equation. This fully discrete scheme only needs to solve a linear system at each time step, and the system is unique solvable. We prove the proposed scheme is unconditionally energy stable. Under a weak regularity hypothesis on the exact solution, we present optimal error estimates for the velocity, magnetic induction and temperature. Finally, several numerical examples are performed to demonstrate both accuracy and efficiency of our proposed scheme.

Weak Regularity of the Inverse Under Minimal Assumptions

Let $$\Omega \subset {\mathbb {R}}^3$$ be a domain and let $$f\in BV_{{\text {loc}}}(\Omega ,{\mathbb {R}}^3)$$ be a homeomorphism such that its distributional adjugate is a finite Radon measure. We show that its inverse has bounded variation $$f^{-1}\in BV_{{\text {loc}}}$$ . The condition that the distributional adjugate is finite measure is not only sufficient but also necessary for the weak regularity of the inverse.

A primal-dual finite element method for first-order transport problems

This article devises a new numerical method for first-order transport problems by using the primal-dual weak Galerkin (PD-WG) finite element method recently developed in scientific computing. The PD-WG method is based on a variational formulation of the modeling equation for which the differential operator is applied to the test function so that low regularity for the exact solution of the original equation is sufficient for computation. The PD-WG finite element method indeed yields a symmetric system involving both the original equation for the primal variable and its dual for the dual variable (also known as Lagrangian multiplier). For the linear transport problem, it is shown that the PD-WG method offers numerical solutions that conserve mass locally on each element. Optimal order error estimates in various norms are derived for the numerical solutions arising from the PD-WG method with weak regularity assumptions on the modelling equations. A variety of numerical results are presented to demonstrate the accuracy and stability of the new method.

New nonasymptotic convergence rates of stochastic proximal point algorithm for stochastic convex optimization

Large sectors of the recent optimization literature focused in the last decade on the development of optimal stochastic first-order schemes for constrained convex models under progressively relaxed assumptions. Stochastic proximal point is an iterative scheme born from the adaptation of proximal point algorithm to noisy stochastic optimization, with a resulting iteration related to stochastic alternating projections. Inspired by the scalability of alternating projection methods, we start from the (linear) regularity assumption, typically used in convex feasiblity problems to guarantee the linear convergence of stochastic alternating projection methods, and analyze a general weak linear regularity condition which facilitates convergence rate boosts in stochastic proximal point schemes. Our applications include many non-strongly convex functions classes often used in machine learning and statistics. Moreover, under weak linear regularity assumption we guarantee convergence rate for SPP, in terms of the distance to the optimal set, using only projections onto a simple component set. Linear convergence is obtained for interpolation setting, when the optimal set of the expected cost is included into the optimal sets of each functional component.

Strict Positivity for the Principal Eigenfunction of Elliptic Operators with Various Boundary Conditions

Abstract We consider elliptic operators with measurable coefficients and Robin boundary conditions on a bounded domain Ω ⊂ ℝ d {\Omega\subset\mathbb{R}^{d}} and show that the first eigenfunction v satisfies v ⁢ ( x ) ≥ δ > 0 {v(x)\geq\delta>0} for all x ∈ Ω ¯ {x\in\overline{\Omega}} , even if the boundary ∂ ⁡ Ω {\partial\Omega} is only Lipschitz continuous. Under such weak regularity assumptions the Hopf–Oleĭnik boundary lemma is not available; instead we use a new approach based on an abstract positivity improving condition for semigroups that map L p ⁢ ( Ω ) {L_{p}(\Omega)} into C ⁢ ( Ω ¯ ) {C(\overline{\Omega})} . The same tool also yields corresponding results for Dirichlet or mixed boundary conditions. Finally, we show that our results can be used to derive strong minimum and maximum principles for parabolic and elliptic equations.

An H2N2 Interpolation for Caputo Derivative with Order in (1, 2) and Its Application to Time-Fractional Wave Equations in More Than One Space Dimension

In this paper, a new derived method is developed for a known numerical differential formula of the Caputo fractional derivative of order $$\gamma \in (1,2)$$ (Li and Zeng in Numerical methods for fractional calculus. Chapman & Hall/CRC numerical analysis and scientific computing, CRC Press, Boca Raton, 2015) by means of the quadratic interpolation polynomials, and a concise expression of the truncation error is given. This new method will be called as the H2N2 method because of the application of the quadratic Hermite and Newton interpolation polynomials. A finite difference scheme with a second order accuracy in space and a $$(3-\gamma )$$-th order accuracy in time based on the H2N2 method is constructed for the initial boundary value problem of time-fractional wave equations. The stability and convergence of the difference scheme are proved. Furthermore, in order to increase computational efficiency, using the sum-of-exponentials to approximate the kernel $$t^{1-\gamma }$$, a fast difference scheme is presented. The problem with weak regularity at the initial time is also discussed with the help of the graded meshes. At each time level, the difference scheme is solved with a fast Poisson solver. Numerical results show the effectiveness of the two difference schemes and confirm our theoretical analysis.