scholarly journals On Weak Super Ricci Flow through Neckpinch

2021 ◽  
Vol 9 (1) ◽  
pp. 120-159
Author(s):  
Sajjad Lakzian ◽  
Michael Munn

Abstract In this article, we study the Ricci flow neckpinch in the context of metric measure spaces. We introduce the notion of a Ricci flow metric measure spacetime and of a weak (refined) super Ricci flow associated to convex cost functions (cost functions which are increasing convex functions of the distance function). Our definition of a weak super Ricci flow is based on the coupled contraction property for suitably defined diffusions on maximal diffusion subspaces. In our main theorem, we show that if a non-degenerate spherical neckpinch can be continued beyond the singular time by a smooth forward evolution then the corresponding Ricci flow metric measure spacetime through the singularity is a weak super Ricci flow for a (and therefore for all) convex cost functions if and only if the single point pinching phenomenon holds at singular times; i.e., if singularities form on a finite number of totally geodesic hypersurfaces of the form {x} × 𝕊 n . We also show the spacetime is a refined weak super Ricci flow if and only if the flow is a smooth Ricci flow with possibly singular final time.

Author(s):  
Peng Lu ◽  
Jiuru Zhou

AbstractWe construct the ancient solutions of the hypersurface flows in Euclidean spaces studied by B. Andrews in 1994.As time {t\rightarrow 0^{-}} the solutions collapse to a round point where 0 is the singular time. But as {t\rightarrow-\infty} the solutions become more and more oval. Near the center the appropriately-rescaled pointed Cheeger–Gromov limits are round cylinder solutions {S^{J}\times\mathbb{R}^{n-J}}, {1\leq J\leq n-1}. These results are the analog of the corresponding results in Ricci flow ({J=n-1}) and mean curvature flow.


2009 ◽  
Vol 23 (03) ◽  
pp. 477-480 ◽  
Author(s):  
ZHILI TANG

The Taguchi robust design concept is combined with the multi-objective deterministic optimization method to overcome single point design problems in Aerodynamics. Starting from a statistical definition of stability, the method finds, Nash equilibrium solutions for performance and its stability simultaneously.


Author(s):  
Татьяна Ароновна Лавина ◽  
Николай Игоревич Степанов

Данная статья посвящена исследованию систем дистанционного обучения в вузах Чувашской Республики, определению основных функций систем управления дистанционным обучением. Проведен анализ систем дистанционного взаимодействия со студентами в федеральных вузах Чувашской Республики. С целью формирования требований к функциям современной единой универсальной платформы дистанционного взаимодействия в университете проведен анализ корпоративной платформы, объединяющей в рабочем пространстве чат, встречи, заметки и вложения MS Teams, позволяющей организовывать дистанционное обучение. На примере веб-ресурсов для поддержки организации дистанционного обучения государственных университетов Чувашской Республики, таких как система управления курсами, использующая лицензию GNU GPL Moodle, системы видеоконференций Big Blue Button, рассматриваются их функциональные возможности и недостатки, выявляются ключевые особенности построения единого информационного пространства вуза на основе идеи использования инструментов образовательных платформ в обеспечении электронной поддержки дистанционного обучения. Представлены преимущества интеграции систем LMS Moodle и MS Teams для построения интерактивного формата обучения на примере построения единой образовательной среды в ЧГПУ им. И. Я. Яковлева. Рассмотрено внедрение единой учетной записи для организации принципа единой точки доступа к информационным системам дистанционного обучения для построения единой электронной образовательной среды университета на базе технологии OpenID Connect. This article is devoted to the study of distance learning systems in universities of the Chuvash Republic, the definition of the main functions of distance learning systems. It presents the analysis of systems of distance learning interaction with students in federal universities of the Chuvash Republic. In order to form the requirements for the functions of a modern unified universal platform for distance learning interaction at the university, the authors analysed the corporate platform that combines chat, meetings, notes and attachments of MS Teams in the workspace. Using the example of web resources to support the organization of distance learning of state universities of the Chuvash Republic, such as the course management system using the GNU GPL Moodle license, the Big Blue Button video conferencing systems, their functionality and shortcomings are considered, the key features of building a unified information space of the university based on the idea of using the tools of educational platforms in providing electronic support for distance learning are identified. The advantages of integrating the LMS Moodle and MS Teams systems for building an interactive learning format are presented on the example of building a unified educational environment at I. Yakovlev CHSPU. The introduction of a single account for the organization of a single point of access to information systems of distance learning for the construction of a single electronic educational environment of the university based on OpenID Connect technology is considered.


2021 ◽  
pp. 2140011
Author(s):  
Tomás Caraballo ◽  
Tran Bao Ngoc ◽  
Tran Ngoc Thach ◽  
Nguyen Huy Tuan

This paper is concerned with the mathematical analysis of terminal value problems (TVP) for a stochastic nonclassical diffusion equation, where the source is assumed to be driven by classical and fractional Brownian motions (fBms). Our two problems are to study in the sense of well-posedness and ill-posedness meanings. Here, a TVP is a problem of determining the statistical properties of the initial data from the final time data. In the case [Formula: see text], where [Formula: see text] is the fractional order of a Laplace operator, we show that these are well-posed under certain assumptions. We state a definition of ill-posedness and obtain the ill-posedness results for the problems when [Formula: see text]. The major analysis tools in this paper are based on properties of stochastic integrals with respect to the fBm.


2019 ◽  
Vol 21 (02) ◽  
pp. 1940010 ◽  
Author(s):  
Pierre Von Mouche ◽  
Takashi Sato

We consider the equilibrium uniqueness problem for a large class of Cournot oligopolies with convex cost functions and a proper price function [Formula: see text] with decreasing price flexibility. This class allows for (at [Formula: see text]) discontinuous industry revenue and in particular for [Formula: see text]. The paper illustrates in an exemplary way the Selten–Szidarovszky technique based on virtual backward reply functions. An algorithm for the calculation of the unique equilibrium is provided.


Author(s):  
Etienne Tang ◽  
Mickaël Philit ◽  
Gilles Leroy ◽  
Isabelle Trebinjac ◽  
Ghislaine Ngo Boum

This paper focuses on an axisymmetric slot injecting cooling air at the casing between the stator and the rotor in a one-stage unshrouded transonic high pressure turbine. This configuration has been studied with the help of unsteady RANS computations with and without the slot. Special care has been taken to model and describe the interaction induced unsteady mechanisms. It has been found that the cooling air is ejected from the axisymmetric slot at a fixed position with respect to the stator vanes, with a much lower incidence angle than the main stream. The flow through the rotor passage is highly modified and reveals an unsteady behaviour which highlights the necessity of using unsteady simulations in order to accurately model such a configuration. The effect on the efficiency and on the repartition of loss generation has been determined. As several different definitions of the efficiency can be used for cooled turbine cases, this choice is discussed. In particular, Young & Horlock’s “Weighted Pressure” definition, which takes into account some unavoidable mixing losses in the definition of the ideal process, is evaluated. With this definition, the slot does not yield any significant decrease in overall efficiency.


Author(s):  
J. H. Horlock ◽  
Leonardo Torbidoni

The efficiency of a cooled turbine stage has been discussed in the literature. All proposed definitions compare the actual power output with an ideal output, which has to be determined; but usually, one of two definitions has been used by turbine designers. In the first, the so-called Hartsel efficiency, the mainstream gas flow, and the various coolant flows to rotor and stator are assumed to expand separately and isentropically to the backpressure. In the second, it is assumed that these flows mix at constant (mainstream) gas pressure before expanding isentropically (sometimes, the rotor coolant flow is ignored in this definition). More recently, it has been suggested that a thermodynamically sounder definition is one in which the gas and coolant flows mix reversibly and adiabatically before isentropic expansion to the backpressure. In the current paper, these three efficiencies are compared, for a typical stage—the first cooled stage of a multistage industrial gas turbine. It is shown that all the efficiencies fall more or less linearly with increase of the fractional (total) coolant flow. It is also shown that the new definition of efficiency gives values considerably lower than the other two efficiencies, which are more widely used at present. Finally, the various irreversibilities associated with the flow through a cooled turbine are calculated. Although all these irreversibilities increase with the fractional coolant flow, it is shown that the “thermal” irreversibility associated with film cooling is higher than the other irreversibilities at large fractional coolant flow.


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