Terms Involving the Divergence Equation

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Small Letter ◽  
Divergence Equation ◽  
Interaction Term ◽  
Transport Term

This chapter estimates the terms in the stress which involve solving a divergence equation of the form ∂ⱼQsuperscript jl = Usuperscript l = esuperscript iGreek Small Letter Lamda Greek Small Letter Xiusuperscript l. These terms are the High–Low Interaction term, the main High–High terms, the remainder of the High–High terms, and the Transport term. For each of these factors, the parametrix expansion for the divergence equation is used. The error of the expansion is eliminated by solving the divergence equation. The chapter also considers the bounds which are obeyed for the parametrices of the oscillatory terms and concludes by applying the parametrix.

Basic Technical Outline

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Vector Field ◽  
Bounded Number ◽  
Divergence Free Vector ◽  
Individual Wave ◽  
Error Terms ◽  
Interaction Term ◽  
Divergence Free ◽  
Complex Valued ◽  
Stress Term ◽  
Transport Term

This chapter provides a more technical outline of the construction, starting with a solution to the Euler-Reynolds system and a correction v₁ = v + V, p₁ = p + P. The correction V is a divergence free vector field which oscillates rapidly compared to v. In the construction, there will always be a bounded number of waves Vsubscript I (at most 192) which are nonzero at any given time t. Each individual wave Vsubscript I composing V is a complex-valued, divergence free vector field that oscillates rapidly in only one direction. The chapter introduces several ways in which to represent each Vsubscript I. Finally, it presents five main error terms: the Transport term, the High–Low Interaction term, the High–High Interference terms, the Stress term, and the Mollification terms.

Stress Terms Not Involving Solving the Divergence Equation

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Energy Levels ◽  
Material Derivative ◽  
Small Letter ◽  
Divergence Equation ◽  
Stress Term

This chapter estimates the terms in the new stress that do not involve solving the divergence equation. These terms are the Mollification terms and the Stress term. Throughout the estimates, Bsubscript Greek Small Letter Lamda will be assumed to be some constant. After considering the Mollification term from the velocity, the chapter introduces a proposition stating that for k = 0, … , L, there exist constants Cₖ depending on Bsubscript Greek Small Letter Lamda. It then estimates the material derivative, highlighting wastefulness in the estimate, and discusses a commutator estimate suggesting that it may be important to work with frequency energy levels of order L greater than or equal to 2. Finally, it presents the Mollification term from the stress as well as estimates for the Stress term.

Frequency and Energy Levels

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Weak Solutions ◽  
Energy Levels ◽  
Integration Scheme ◽  
Hölder Regularity ◽  
Reynolds Equations ◽  
Integration Procedure ◽  
Convex Integration ◽  
Interaction Term ◽  
Holder Regularity ◽  
Transport Term

This chapter shows how to measure the Hölder regularity of the weak solutions that are constructed when the scheme is executed more carefully. For this aspect of the convex integration scheme, a notion of frequency energy levels is introduced. This notion is meant to accurately record the bounds which apply to the (v, p, R) coming from the previous stage of the construction. The chapter presents an example of a candidate definition for frequency and energy levels. Based on this definition, the effect of one iteration of the convex integration procedure can be summarized in a single lemma, which states that there is a solution to the Euler-Reynolds equations with new frequency and energy levels. The chapter also considers the High–Low Interaction term and the Transport term.

The Coarse Scale Velocity

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Velocity Field ◽  
Building Blocks ◽  
Coarse Scale ◽  
Divergence Equation ◽  
Higher Derivatives ◽  
Order Of Magnitude ◽  
Derivatives Of

This chapter deals with the coarse scale velocity. It begins the proof of Lemma (10.1) by choosing a double mollification for the velocity field. Here ∈ᵥ is taken to be as large as possible so that higher derivatives of velement are less costly, and each vsubscript Element has frequency smaller than λ‎ so elementv⁻¹ must be smaller than λ‎ in order of magnitude. Each derivative of vsubscript Element up to order L costs a factor of Ξ‎. The chapter proceeds by describing the basic building blocks of the construction, the choice of elementv and the parametrix expansion for the divergence equation.

Does Enhancing Mind Perception Affect Conspiracy Belief?

2019 ◽  
Author(s):  
Jorge Noguera
Keyword(s):  
United States ◽  
Southeastern United States ◽  
Social Discourse ◽  
Psychological Processes ◽  
Social Psychological ◽  
Conspiracy Theories ◽  
Mind Perception ◽  
Mixed Anova ◽  
Interaction Term ◽  
The Mind

This study was conducted to determine the effectiveness of a novel mind perception manipulation. Mind perception is currently theorized to be an essential aspect of a number of human social psychological processes. Thus, a successful manipulation would allow for the causal study of those processes. This manipulation was created in an attempt to explore the downstream impact of mind perception on the endorsement of conspiracy theories. Conspiracy theories are steadily becoming more and more prominent in social discourse. Endorsement of conspiracy theories are beginning to show real world ramifications such as a danger to human health (e.g., in the anti-vaccination movement). A sample of college students (valid N = 53) from a large rural institution in the southeastern United States participated for course credit. These participants completed a mind perception pretest, were randomly assigned to either the manipulation in question (in which participants are asked to consider the ‘mind’ of several targets and write their thoughts about them) or the control condition, and then they completed a posttest. The mixed ANOVA revealed that the interaction term between Time and Condition was not significant. Because the manipulation did not work, other analyses were aborted, in accord with the pre-registration. My Discussion focuses on the procedures and potential shortcomings of this manipulation, in an effort to lay the groundwork for a successful one.

scholarly journals A Cortical-Inspired Sub-Riemannian Model for Poggendorff-Type Visual Illusions

2021 ◽  
Vol 7 (3) ◽  
pp. 41
Author(s):  
Emre Baspinar ◽  
Luca Calatroni ◽  
Valentina Franceschi ◽  
Dario Prandi
Keyword(s):  
Heat Kernel ◽  
Mathematical Description ◽  
Gradient Descent ◽  
Numerical Results ◽  
Visual Illusions ◽  
Efficient Computation ◽  
Functional Architecture ◽  
Descent Algorithms ◽  
Interaction Term

We consider Wilson-Cowan-type models for the mathematical description of orientation-dependent Poggendorff-like illusions. Our modelling improves two previously proposed cortical-inspired approaches, embedding the sub-Riemannian heat kernel into the neuronal interaction term, in agreement with the intrinsically anisotropic functional architecture of V1 based on both local and lateral connections. For the numerical realisation of both models, we consider standard gradient descent algorithms combined with Fourier-based approaches for the efficient computation of the sub-Laplacian evolution. Our numerical results show that the use of the sub-Riemannian kernel allows us to reproduce numerically visual misperceptions and inpainting-type biases in a stronger way in comparison with the previous approaches.

MODELLING NITROGEN PROCESSES IN SOIL: MATHEMATICAL DEVELOPMENT AND RELATIONSHIPS

1976 ◽  
Vol 56 (2) ◽  
pp. 71-78 ◽  
Author(s):  
D. R. CAMERON ◽  
C. G. KOWALENKO
Keyword(s):  
Time Dependent ◽  
Rate Coefficients ◽  
Nitrification Rate ◽  
Order Kinetic ◽  
Nitrogen Flow ◽  
First Order ◽  
Interaction Term ◽  
Flow Pathways ◽  
Adsorption Desorption ◽  
Temperature Water

A small subsystem model was developed to simulate the major nitrogen flow pathways in an unsaturated soil treated with ammonium sulphate. A nonlinear Freundlich equilibrium model and a Langmuir kinetic model were used to describe mathematically the adsorption–desorption of soluble NH4+ to the exchangeable and clay-fixed phases, respectively. Time dependent, microbial mediated first-order kinetic models were used to quantify the ammonification and nitrification processes. The subsystem model was then used as a research tool to derive ammonification and nitrification rate coefficients for a preceding incubation experiment conducted using different soil moisture contents and temperatures. The model yields reasonably good fits to the observed data. A subsequent regression analysis relating the coefficients to temperature and moisture pointed out the importance of the temperature–water content interaction term in quantifying microbial mediated processes.

scholarly journals 312 Female sex independently predicts adjuvant immunotherapeutic benefit from CTLA4 immune checkpoint inhibition

2020 ◽  
Vol 8 (Suppl 3) ◽  
pp. A339-A339
Author(s):  
Ahmad Tarhini ◽  
Ni Kang ◽  
Sandra Lee ◽  
F Stephen Hodi ◽  
Gary Cohen ◽  
...  
Keyword(s):  
Overall Survival ◽  
Treatment Efficacy ◽  
Significant Interaction ◽  
Cox Regression ◽  
Performance Status ◽  
Phase Iii ◽  
High Dose ◽  
List Type ◽  
Female Sex ◽  
Interaction Term

BackgroundSex differences in tumor immunity and response to immunotherapy were shown in murine models and descriptive analyses from recent clinical trials. Female sex hormones have been implicated in melanoma development and response to systemic therapy. We hypothesized a gender difference in response to adjuvant immunotherapy with ipilimumab (3 or 10 mg/kg; ipi3 or ipi10) versus high dose IFNα (HDI) as tested in the E1609 trial.MethodsE1609 demonstrated significant overall survival (OS) benefit with ipi3 versus HDI.1 We investigated treatment efficacy between ipi and HDI in the subgroups by sex (female, male), age (< 55 or ≥55), stage at study entry (IIIB, IIIC, M1a/1b), ECOG performance status (PS 0, 1), ulceration (yes, no), primary tumor (known, unknown), number of lymph nodes involved (0, 1, 2–3, 4+). Forest plots were created to compare OS and RFS with ipi3 vs. HDI and ipi10 vs. HDI using the concurrently randomized ITT populations. For the estimated HRs, 95% confidence intervals were created for all subgroups.ResultsThe subgroups of female, stage IIIC, PS=1, ulcerated, in-transit without lymph node involvement demonstrated significant improvement in overall survival (OS) and/or relapse free survival (RFS) with ipi3 versus HDI as summarized in table 1. Female sex was significant for both OS and RFS and was further explored. In investigating RFS with ipi3 versus HDI, a multivariate Cox regression model including sex, treatment and interaction term of sex*treatment, indicated a significant interaction between sex and treatment (P = 0.026). Including sex, PS (0 vs. 1), age (<55 vs. 55+), ulceration (yes vs. no), stage (IIIB, IIIC, M1a, M1b), treatment and interaction term of sex*treatment, indicated a significant interaction between sex and treatment (P = 0.024). While similar trends were seen, no significant interactions between sex and treatment effect were found in the OS multivariate analysis or in the comparison of ipi10 versus HDI. When exploring age, in the univariate analyses in the ipi3 versus HDI comparison older women appeared to drive most of the difference (age ≥55: OS, P=0.02 and RFS, P=0.08; differences non-significant for women <55). Table 1.Abstract 312 Table 1Treatment efficacy between ipi3 and HDI by subgroupConclusionsFemale sex was independently associated with RFS adjuvant immunotherapeutic benefit from ipi3, supporting a potentially important role for female related factors in the immune response against melanoma, and these warrant further investigation.Trial RegistrationNCT01274338Ethics ApprovalThe study protocol was approved by the institutional review board (IRB) of each participating institution and conducted in accordance with Good Clinical Practice guidelines as defined by the International Conference on Harmonisation. This study was monitored by the ECOG-ACRIN DataSafety Monitoring Committee and the NCI.ConsentAll patients provided IRB-approved written informed consent.ReferenceTarhini AA, Lee SJ, Hodi FS, Rao UNM, Cohen GI, Hamid O, Hutchins LF, Sosman JA, Kluger HM, Eroglu Z, Koon HB, Lawrence DP, Kendra KL, Minor DR, Lee CB, Albertini MR, Flaherty LE, Petrella TM, Streicher H, Sondak VK, Kirkwood JM. Phase III Study of Adjuvant Ipilimumab (3 or 10 mg/kg) Versus High-Dose Interferon Alfa-2b for Resected High-Risk Melanoma: North American Intergroup E1609. J Clin Oncol. 2020 Feb 20;38(6):567–575. PMID: 31880964.

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