• Kyungpook Mathematical Journal
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• 제45권4호
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• pp.579-594
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• 2005

In this paper, we will establish some new oscillation criteria for the second-order half-linear delay difference equation $${\Delta}(Pn ({\Delta}Xn)^{\gamma})+q_nx_\array{{\gamma}\\n-{\sigma}}=0,\;n{\geq}n_0$$, where ${\gamma}>0$ is a quotient of odd positive integers. Our results in this paper are sharp and improve some of the well known oscillation results in the literature. Some examples are considered to illustrate our main results.

• Journal of applied mathematics & informatics
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• 제36권3_4호
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• pp.213-229
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• 2018

In this work, we establish oscillation of the second order sublinear neutral delay dynamic equations of the form:$$(r(t)((x(t)+p(t)x({\tau}(t)))^{\Delta})^{\gamma})^{\Delta}+q(t)x^{\gamma}({\alpha}(t))+v(t)x^{\gamma}({\eta}(t))=0$$ on a time scale T by means of Riccati transformation technique, under the assumptions $${\displaystyle\smashmargin{2}{\int\nolimits^{\infty}}_{t_0}}\({\frac{1}{r(t)}}\)^{\frac{1}{\gamma}}{\Delta}t={\infty}$$, and ${\displaystyle\smashmargin{2}{\int\nolimits^{\infty}}_{t_0}}\({\frac{1}{r(t)}}\)^{\frac{1}{\gamma}}{\Delta}t$ < ${\infty}$, for various ranges of p(t), where 0 < ${\gamma}{\leq}1$ is a quotient of odd positive integers.

• Journal of applied mathematics & informatics
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• 제30권5_6호
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• pp.1005-1021
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• 2012

This work is concerned with oscillation of the second order sublinear neutral delay dynamic equations of the form $$\(r(t)\;\((y(t)+p(t)y(a(t)))^{\Delta}\)^{\gamma}\)^{\Delta}+q(t)y^{\gamma}({\beta}(t))=0$$ on a time scale $\mathcal{T}$ by means of Riccati transformation technique, under the assumptions $\int^{\infty}_{t_0}\(\frac{1}{r(t)}\)^{\frac{1}{\gamma}}$ ${\Delta}t={\infty}$ and $\int^{\infty}_{t_0}\(\frac{1}{r(t)}\)^{\frac{1}{\gamma}}$ ${\Delta}t$ < ${\infty}$, where 0 < ${\gamma}{\leq}1$ is a quotient of odd positive integers.

• Journal of applied mathematics & informatics
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• 제28권5_6호
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• pp.1285-1304
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• 2010

The purpose of this paper is to establish some sufficient conditions for oscillation of solutions of the second-order functional dynamic equation of Emden-Fowler type $\[a(t)x^{\Delta}(t)\]^{\Delta}+p(t)|x^{\gamma}(\tau(t))|\|x^{\Delta}(t)\|^{1-\gamma}$ $sgnx(\tau(t))=0$, $t\;{\geq}\;t_0$, on a time scale $\mathbb{T}$, where ${\gamma}\;{\in}\;(0,\;1]$, a, p and $\tau$ are positive rd-continuous functions defined on $\mathbb{T}$, and $lim_{t{\rightarrow}{\infty}}\;{\tau}(t)\;=\;\infty$. Our results include some previously obtained results for differential equations when $\mathbb{T}=\mathbb{R}$. When $\mathbb{T}=\mathbb{N}$ and $\mathbb{T}=q^{\mathbb{N}_0}=\{q^t\;:\;t\;{\in}\;\mathbb{N}_0\}$ where q > 1, the results are essentially new for difference and q-difference equations and can be applied on different types of time scales. Some examples are worked out to demonstrate the main results.

• Journal of applied mathematics & informatics
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• 제30권1_2호
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• pp.219-234
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• 2012

By using the generalized Riccati transformation and the inequality technique, we establish some new oscillation criterion for the second-order nonlinear delay dynamic equations $$(a(t)(x^{\Delta}(t))^{\gamma})^{\Delta}+q(t)f(x({\tau}(t)))=0$$ on a time scale $\mathbb{T}$, here ${\gamma}{\geq}1$ is the ratio of two positive odd integers with $a$ and $q$ real-valued positive right-dense continuous functions defined on $\mathbb{T}$. Our results not only extend and improve some known results, but also unify the oscillation of the second-order nonlinear delay differential equation and the second-order nonlinear delay difference equation.

• 대한수학회지
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• 제49권5호
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• pp.1017-1030
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• 2012

We consider forced second order differential equation with $p$-Laplacian and nonlinearities given by a Riemann-Stieltjes integrals in the form of $$(p(t){\phi}_{\gamma}(x^{\prime}(t)))^{\prime}+q_0(t){\phi}_{\gamma}(x(t))+{\int}^b_0q(t,s){\phi}_{{\alpha}(s)}(x(t))d{\zeta}(s)=e(t)$$, where ${\phi}_{\alpha}(u):={\mid}u{\mid}^{\alpha}\;sgn\;u$, ${\gamma}$, $b{\in}(0,{\infty})$, ${\alpha}{\in}C[0,b)$ is strictly increasing such that $0{\leq}{\alpha}(0)<{\gamma}<{\alpha}(b-)$, $p$, $q_0$, $e{\in}C([t_0,{\infty}),{\mathbb{R}})$ with $p(t)>0$ on $[t_0,{\infty})$, $q{\in}C([0,{\infty}){\times}[0,b))$, and ${\zeta}:[0,b){\rightarrow}{\mathbb{R}}$ is nondecreasing. Interval oscillation criteria of the El-Sayed type and the Kong type are obtained. These criteria are further extended to equations with deviating arguments. As special cases, our work generalizes, unifies, and improves many existing results in the literature.

• Korean Journal of Mathematics
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• 제19권4호
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• pp.391-402
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• 2011

We give necessary and sufficient conditions such that the homogeneous differential equations of the type: $$(r(t)x^{\prime}(t))^{\prime}+q(t)x^{\prime}(t)+p(t)x(t)=0$$ are nonoscillatory where $r(t)$ > 0 for $t{\in}I=[{\alpha},{\infty})$, ${\alpha}$ > 0. Under the suitable conditions we show that the above equation is nonoscillatory if and only if for ${\gamma}$ > 0, $$(r(t)x^{\prime}(t))^{\prime}+q(t)x^{\prime}(t)+p(t)x(t-{\gamma})=0$$ is nonoscillatory. We obtain several comparison theorems.

• Journal of Yeungnam Medical Science
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• 제39권4호
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• pp.269-277
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• 2022

The amyloid hypothesis has been considered a major explanation of the pathogenesis of Alzheimer disease. However, failure of phase III clinical trials with anti-amyloid-beta monoclonal antibodies reveals the need for other therapeutic approaches to treat Alzheimer disease. Compared to its relatively short history, optogenetics has developed considerably. The expression of microbial opsins in cells using genetic engineering allows specific control of cell signals or molecules. The application of optogenetics to Alzheimer disease research or clinical approaches is increasing. When applied with gamma entrainment, optogenetic neuromodulation can improve Alzheimer disease symptoms. Although safety problems exist with optogenetics such as the use of viral vectors, this technique has great potential for use in Alzheimer disease. In this paper, we review the historical applications of optogenetic neuromodulation with gamma entrainment to investigate the mechanisms involved in Alzheimer disease and potential therapeutic strategies.

• Kyungpook Mathematical Journal
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• 제56권1호
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• pp.137-146
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• 2016

In this paper we study the oscillation criteria for the second order nonlinear differential equation with delay and advanced arguments of the form $$([x(t)+a(t)x(t-{\sigma}_1)+b(t)x(t+{\sigma}_2)]^{\alpha})^{{\prime}{\prime}}+q(t)x^{\beta}(t-{\tau}_1)+q(t)x^{\gamma}(t+{\tau}_2)=0,\;t{\geq}t_0$$ where ${\sigma}_1$, ${\sigma}_2$, ${\tau}_1$ and ${\tau}_2$ are nonnegative constants and ${\alpha}$, ${\beta}$ and ${\gamma}$ are the ratios of odd positive integers. Examples are provided to illustrate the main results.

• 생물정신의학
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• 제8권2호
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• pp.251-257
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• 2001

Backgrounds : Gamma band oscillatory activity is considered to be related to cognitive functions and illustrates that the concept of event-related oscillations bridges the gap between single neurons and neural assemblies. An event-related gamma oscillation is the time-locked responses of specific frequency, and can be identified by computing the amplitude frequency characteristics of the averaged event-related potentials(ERPs) after stimulation. Objectives : We purposed to present experimental paradigm to investigate ${\gamma}$-band oscillation activities from the recording of ERPs by using auditory oddball paradigm and investigate the difference of ${\gamma}$-band activity between schizophrenia and normal controls. Methods : The ERPs resulting from auditory stimuli with oddball paradigm in a group of schizophrenics(n=11), and also a group of age-, sex-, and handedness matched normal controls, were recorded by 128 channel EEG. The ${\gamma}$-band oscillatory activities were calculated by using time-frequency wavelet decomposition of the signal between 20 and 80Hz. The ${\gamma}$-band oscillatory activities of both groups were compared by t-test. Results : The ${\gamma}$-band oscillatory of the leads Fz, Cz, and Pz of both groups were represented well in the time-frequency maps. Significant increases of the ${\gamma}$-band activity in normal controls compared with schizophrenics were observed around 160 msec, 350 msec, and 800 msec after stimulation. Conclusions : Our results suggested that the increment in ${\gamma}$-band oscillatory activity during cognitive operations and decreased ${\gamma}$-band activity in schizophrenics may be associated with the cognitive dysfunctions and the pathophysiology of the schizophrenia.