stationary function
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2021 ◽  
Author(s):  
Samarth Motagi ◽  
Sirish Namilae ◽  
Audrey Gbaguidi ◽  
Scott Parr ◽  
Dahai Liu

Abstract Secondary crashes or crashes that occur in the wake of a preceding or primary crash are among the most critical incidents occurring on highways, due to the exceptional danger they present to the first responders and victims of the primary crash. In this work, we developed a self-exciting temporal point process to analyze crash events data and classify it into primary and secondary crashes. Our model uses a self-exciting function to describe secondary crashes while primary crashes are modeled using a background rate function. We fit the model to crash incidents data from the Florida Department of Transportation, on Interstate-4 (I-4) highway for the years 2015-2017, to determine the model parameters. These are used to estimate the probability that a given crash is secondary crash and to find queue times. To represent the periodically varying traffic levels and crash incidents, we model the background rate, as a stationary function, a sinusoidal non-stationary function, and a piecewise non-stationary function. We show that the sinusoidal non-stationary background rate fits the traffic data better and replicates the daily and weekly peaks in crash events due to traffic rush hours. Secondary crashes are found to account for up to 15.09% of the traffic incident, depending on the city on the I4 Highway.


2020 ◽  
pp. 6-11
Author(s):  
A.M. Aliyev ◽  
◽  
A.R. Safarov ◽  
I.V. Balayev ◽  
I.I. Osmanova ◽  
...  

The mathematical model has been developed for the process polymerization propylene proceeding under unsteady conditions due to the toxic effect of methylacetylene on it, leading to decrease of the productivity and quality of polypropylene. Non-stationary function to maintain the productivity at the optimum level obtained during the process in stationary conditions has been proposed. Using this mathematical model allow ones control the process, stabilize it at any time of the polymerization operation. The control scheme of algorithm of this process has been created


Author(s):  
А.В. Сергеевичев ◽  
А.М. Артеменков

Актуальность проблемы получения необходимого качества обработанной поверхности древесины в настоящее время становится все более значимой в связи с повышением требований к качеству продукции деревообработки и получения наибольшего экономического эффекта для предприятия. Вопросом математического моделирования поверхности шлифовального инструмента с целью объективного описания его рельефа занимались многие исследователи. Так, при решении задачи установления взаимосвязи рельефа инструмента с шероховатостью поверхности обрабатываемой детали использовались методы представления его в виде набора профилей: в виде «резьбовой гребенки», в виде реализации случайной стационарной функции, в виде элементарных эффективных профилей. Второй путь моделирования поверхности инструмента, использовавшийся при установлении связи между характеристиками рельефа и его режущей способностью, заключается в представлении абразивных зерен в виде тел различной формы: шара, эллипсоида. Достаточно популярна статистическая многогранная модель абразивного зерна. По современным представлениям, основной причиной снижения работоспособности и износостойкости шлифовальных шкурок при обработке древесины является засаливание, которое происходит вследствие защемления сошлифованных частиц древесины в межзерновом пространстве, прилипания частиц шлифовальной пыли к поверхности шкурки, что обусловлено липкостью смазки и выделением смолистых веществ древесины при нагреве, накоплением на шкурке электростатического заряда, недостаточным объемом свободного межзернового пространства. Последовательное повышение давления прижима позволяет более полно использовать режущую способность шлифовальной ленты. Анализ показывает, что длительность каждого последующего этапа увеличения прижима будет больше, чем предыдущего, в связи с уменьшением доли застрявшей древесины, в то время как производительность процесса шлифования остается на прежнем уровне. Для увеличения износостойкости шлифовальных лент необходимо увеличивать объем межзернового пространства и уменьшать объем застревающей в этом пространстве стружки. Relevance of a problem of obtaining necessary quality of the processed surface of wood becomes more and more significant in connection with increase in requirements to quality of production of a woodworking and obtaining the greatest economic effect for the enterprise now. Many researchers dealt with an issue of mathematical model operation of a surface of the grinding tool for the purpose of the objective description of its relief. So, at the solution of a problem of establishment of interrelation of a relief of the tool with a roughness of a surface of the processed detail methods of its representation in the form of a set of profiles were used: in the form of "a carving comb", in the form of realization of random stationary function, in the form of the partial efficient profiles. The second path of model operation of a surface of the tool used at establishment of communication between characteristics of a relief and its cutting ability consists in representation of abrasive grains in the form of bodies of various forms: sphere, ellipsoid. The statistical polyhedral model of abrasive grain is rather popular. On the modern representations, the main reason for decrease in serviceability and wear resistance of abrasive papers when processing wood is the balling which occurs owing to jamming the unbound particles of wood in intergranular space, sticking of particles of grinding dust to a skin surface that is caused by stickiness of lubricant and selection of resinous substances of wood when heating, accumulation on a skin of an electrostatic charge, the poor volume of the free intergranular space. Serial build-up of pressure of a clip allows to use the cutting ability of an abrasive belt more fully. The analysis shows that duration of each subsequent stage of increase in a clip will be more, than previous, in connection with decrease of a share of the got stuck wood while efficiency of process of grinding remains at the previous level. For increase in wear resistance of abrasive belts it is necessary to increase the volume of intergranular space and to reduce the volume of the shaving which is getting stuck in this space.


2012 ◽  
Vol 10 (6) ◽  
Author(s):  
Małgorzata Klimek ◽  
Marek Błasik

AbstractTwo-term semi-linear and two-term nonlinear fractional differential equations (FDEs) with sequential Caputo derivatives are considered. A unique continuous solution is derived using the equivalent norms/metrics method and the Banach theorem on a fixed point. Both, the unique general solution connected to the stationary function of the highest order derivative and the unique particular solution generated by the initial value problem, are explicitly constructed and proven to exist in an arbitrary interval, provided the nonlinear terms fulfil the corresponding Lipschitz condition. The existence-uniqueness results are given for an arbitrary order of the FDE and an arbitrary partition of orders between the components of sequential derivatives.


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