Weak solutions and the universal iterative process

1995 ◽  
pp. 1-22
Author(s):  
Alexander Koshelev
Keyword(s):  
Weak Solutions ◽  
Iterative Process

Bayesian Applications in Auditory Research

2019 ◽  
Vol 62 (3) ◽  
pp. 577-586 ◽  
Author(s):  
Garnett P. McMillan ◽  
John B. Cannon
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Bayesian Methods ◽  
Prior Distribution ◽  
Interim Analysis ◽  
Iterative Process ◽  
Bayes Theorem ◽  
Sound Therapy ◽  
The Many ◽  
Auditory Research

Purpose This article presents a basic exploration of Bayesian inference to inform researchers unfamiliar to this type of analysis of the many advantages this readily available approach provides. Method First, we demonstrate the development of Bayes' theorem, the cornerstone of Bayesian statistics, into an iterative process of updating priors. Working with a few assumptions, including normalcy and conjugacy of prior distribution, we express how one would calculate the posterior distribution using the prior distribution and the likelihood of the parameter. Next, we move to an example in auditory research by considering the effect of sound therapy for reducing the perceived loudness of tinnitus. In this case, as well as most real-world settings, we turn to Markov chain simulations because the assumptions allowing for easy calculations no longer hold. Using Markov chain Monte Carlo methods, we can illustrate several analysis solutions given by a straightforward Bayesian approach. Conclusion Bayesian methods are widely applicable and can help scientists overcome analysis problems, including how to include existing information, run interim analysis, achieve consensus through measurement, and, most importantly, interpret results correctly. Supplemental Material https://doi.org/10.23641/asha.7822592

Identifying changes in trends in the age standardized incidence of melanoma in Australia

2019 ◽  
Vol 3 (4) ◽  
pp. 250-252 ◽  
Author(s):  
David M Hille
Keyword(s):  
Public Health ◽  
Linear Regression ◽  
Iterative Process ◽  
Linear Trend ◽  
Piecewise Linear ◽  
Health Issue ◽  
Public Health Issue ◽  
Piecewise Linear Regression ◽  
Significant Public Health ◽  
Males And Females

ObjectiveTo identify changes in the linear trend of the age-standardized incidence of melanoma in Australia for all persons, males, and females. MethodsA two-piece piecewise linear regression was fitted to the data. The piecewise breakpoint varied through an iterative process to determine the model that best fits the data.ResultsStatistically significant changes in the trendof the age-standardized incidence of melanoma in Australia were found for all persons, males, and females. The optimal breakpoint for all persons and males was at 1998. For females, the optimal breakpoint was at 2005. The trend after these breakpoints was flatter than prior to the breakpoints, but still positive.ConclusionMelanoma is a significant public health issue in Australia. Overall incidence continues to increase. However, the rate at which the incidence is increasing appears to be decreasing.

DESIGN APPROACH TO REHABILITATION: DEVELOPING THERAPY ASSISTIVE PRODUCTS FOR CHILDREN WITH HEMIPLEGIC CEREBRAL PALSY

2018 ◽  
Vol 12 (2) ◽  
pp. 307
Author(s):  
G.Y.A. Shanya I. Perera ◽  
W.M.N. Dilshani Ranasinghe
Keyword(s):  
Cerebral Palsy ◽  
Functional Outcomes ◽  
Iterative Process ◽  
Participatory Design ◽  
Design Methodology ◽  
Physical Disabilities ◽  
Comprehensive Understanding ◽  
Constructive Feedback ◽  
Children With Disability ◽  
Hemiplegic Cerebral Palsy

Therapy plays an important role in rehabilitation of children suffering from physical disabilities. Disability conditions like Hemiplegic Cerebral Palsy require vigorous therapy measures, which could be unappealing to children. Using therapy assistive products for rehabilitation can make therapy activities engaging and appealing to children and yield effective outcomes. However, there is limited availability of context based therapy assistive products, which are engaging, and appealing to children suffering from Hemiplegic Cerebral Palsy. This study explores how design methodology can be used to develop therapy assistive products for rehabilitation of children with disability. The study is based on developing a set of therapy assistive products to improve the hand-skills of children with Hemiplegic Cerebral Palsy. Developing therapy assistive products require comprehensive understanding of therapeutic aspects, design aspects and careful integration of the two disciplines. Hence, practicing multidisciplinary and participatory design approaches in the design process is imperative. Usability of therapy assistive products are highly impactive in nature, and therefore an iterative process of prototyping, testing, receiving constructive feedback and developing the products based on feedback should be adopted to achieve feasible and  functional outcomes.

scholarly journals Local versus nonlocal elliptic equations: short-long range field interactions

2020 ◽  
Vol 10 (1) ◽  
pp. 895-921
Author(s):  
Daniele Cassani ◽  
Luca Vilasi ◽  
Youjun Wang
Keyword(s):  
Asymptotic Behavior ◽  
Maximum Principle ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Spectral Properties ◽  
Fractional Laplacian ◽  
Coupling Parameter ◽  
Nonlocal Operators ◽  
The Maximum Principle ◽  
Regularity Results

Abstract In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter vanishes (which turns the problem into a purely nonlocal one) or goes to infinity (reducing the problem to the classical semilinear Laplace equation) is also investigated.

scholarly journals Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

2021 ◽  
Author(s):  
Shohei Nakajima
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Stochastic Partial Differential Equations ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Partial Differential ◽  
Continuity Theorem ◽  
Existence Of Weak Solutions ◽  
One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

scholarly journals Equivalence of viscosity and weak solutions for a p-parabolic equation

2021 ◽  
Author(s):  
Jarkko Siltakoski
Keyword(s):  
Parabolic Equation ◽  
Weak Solutions ◽  
Lower Semicontinuity ◽  
Lower Semicontinuous ◽  
Relationship Of ◽  
The Relationship

AbstractWe study the relationship of viscosity and weak solutions to the equation $$\begin{aligned} \smash {\partial _{t}u-\varDelta _{p}u=f(Du)}, \end{aligned}$$ ∂ t u - Δ p u = f ( D u ) , where $$p>1$$ p > 1 and $$f\in C({\mathbb {R}}^{N})$$ f ∈ C ( R N ) satisfies suitable assumptions. Our main result is that bounded viscosity supersolutions coincide with bounded lower semicontinuous weak supersolutions. Moreover, we prove the lower semicontinuity of weak supersolutions when $$p\ge 2$$ p ≥ 2 .

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