• Journal of Applied and Pure Mathematics
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• v.4 no.3_4
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• pp.141-150
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• 2022

The aim of this article is to introduce hesitant fuzzy paraopen and hesitant fuzzy mean open sets in hesitant fuzzy topological spaces. Moreover we investigate and extend some properties of hesitant fuzzy mean open with hesitant fuzzy paraopen, hesitant fuzzy minmimal open and maximal open sets in hesitant fuzzy topological spaces.

• Journal of the Korean Mathematical Society
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• v.60 no.2
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• pp.255-271
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• 2023

In this paper, we deal with random attractors for dynamical systems forced by a deterministic noise. These kind of systems are modeled as skew products where the dynamics of the forcing process are described by the base transformation. Here, we consider skew products over the Bernoulli shift with the unit interval fiber. We study the geometric structure of maximal attractors, the orbit stability and stability of mixing of these skew products under random perturbations of the fiber maps. We show that there exists an open set U in the space of such skew products so that any skew product belonging to this set admits an attractor which is either a continuous invariant graph or a bony graph attractor. These skew products have negative fiber Lyapunov exponents and their fiber maps are non-uniformly contracting, hence the non-uniform contraction rates are measured by Lyapnnov exponents. Furthermore, each skew product of U admits an invariant ergodic measure whose support is contained in that attractor. Additionally, we show that the invariant measure for the perturbed system is continuous in the Hutchinson metric.

• Journal of the Korean Mathematical Society
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• v.36 no.4
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• pp.787-811
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• 1999

Let M be the maximal ideal space of the Banach algebra $H^{\infty}$ of bounded analytic functions on the open unit disc $\triangle$. For a positive singular measure ${\mu}\;on\;{\partial\triangle},\;let\;{L_{+}}^1(\mu)$ be the set of measures v with $0\;{\leq}\;{\nu}\;{\ll}\;{\mu}\;and\;{{\psi}_{\nu}}$ the associated singular inner functions. Let $R(\mu)\;and\;R_0(\mu)$ be the union sets of $\{$\mid$\psiv$\mid$\;<\;1\}\;and\;\{$\mid${\psi}_{\nu}$\mid$\;<\;0\}\;in\;M\;{\setminus}\;{\triangle},\;{\nu}\;\in\;{L_{+}}^1(\mu)$, respectively. It is proved that if $S(\mu)\;=\;{\partial\triangle}$, where $S(\mu)$ is the closed support set of $\mu$, then $R(\mu)\;=\;R0(\mu)\;=\;M{\setminus}({\triangle}\;{\cup}\;M(L^{\infty}(\partial\triangle)))$ is generated by $H^{\infty}\;and\;\overline{\psi_{\nu}},\;{\nu}\;{\in}\;{L_1}^{+}(\mu)$. It is proved that %d{\theta}(S(\mu))\;=\;0$ if and only if there exists as Blaschke product b with zeros $\{Zn\}_n$ such that $R(\mu)\;{\subset}\;{$\mid$b$\mid$\;<\;1}\;and\;S(\mu)$ coincides with the set of cluster points of $\{Zn\}_n$. While, we proved that $\mu$ is a sum of finitely many point measure such that $R(\mu)\;{\subset}\;\{$\mid${\psi}_{\lambda}$\mid$\;<\;1}\;and\;S(\lambda)\;=\;S(\mu)$. Also it is studied conditions on \mu for which $R(\mu)\;=\;R0(\mu)$.

• Transactions of the Korean Society of Automotive Engineers
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• v.19 no.2
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• pp.50-56
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• 2011

To meet the NOx limit without a penalty of fuel consumption, Urea-SCR system is currently regarded as promising NOx reduction technology for diesel engines. SCR system has to achieve maximal NOx conversion in combination with minimal $NH_3$ slip. In this study, map based open loop control for urea injection was developed and assessed in the European Transient Cycle (ETC) for heavy duty diesel engine. The basic urea quantity set-value which was calculated using the look up tables of engine out NOx, exhaust flow rate and optimum NSR resulted in NOx reduction of 80% and the average $NH_3$ slip of 24 ppm and maximum of 79 ppm. In order to reduce $NH_3$ slip, $NH_3$ storage control algorithm was applied to correct the basic urea quantity and reduced $NH_3$ slip levels to the average 15 ppm and maximum 49 ppm while keeping NOx reduction of 76%. With high and increasing SCR temperature, the $NH_3$ storage capacity decreases, which leads to $NH_3$ slip. The resulting $NH_3$ slip peak can be avoided by stopping or significantly reducing the urea injection during the SCR temperature gradient is over $30^{\circ}C/min$.