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On the Pricing Formula for the Perpetual American Volatility Option Under the Mean-reverting Processes
https://ah.nccu.edu.tw/handle/140.119/137566
title: On the Pricing Formula for the Perpetual American Volatility Option Under the Mean-reverting Processes abstract: This paper studies the properties of the parabolic free-boundary problem arising from pricing of American volatility options in mean-reverting volatility processes. When the volatility index follows the mean-reverting square root process (MRSRP), we derive a closed-form pricing formula for the perpetual American power volatility option. Moreover, an artificial neural network (ANN) approach is extended to find an approximate solution of the free boundary problem arising from pricing the perpetual American option. The comparison results demonstrates that the ANN provides an accurate approach to approximate solution for the free boundary problem.
<br>Traveling waves for a three-component reaction-diffusion model of farmers and hunter-gatherers in the Neolithic transition
https://ah.nccu.edu.tw/handle/140.119/137565
title: Traveling waves for a three-component reaction-diffusion model of farmers and hunter-gatherers in the Neolithic transition abstract: The Neolithic transition began the spread of early agriculture throughout Europe through interactions between farmers and hunter-gatherers about 10,000 years ago. Archeological evidences indicate that the expanding velocity of farming into a region occupied by hunter-gatherers is roughly constant all over Europe. In the late twentieth century, from the contribution of the radiocarbon dating, it could be found that there are two types of farmers: one is the original farmer and the other is the converted farmer which is genetically hunter-gatherers but learned agriculture from neighbouring farmers. Then this raises the following questions: Which farming populations play a key role in the expansion of farmer populations in Europe? and what is the fate of hunter-gatherers (e.g., become extinct, or live in lower density, or live in agricultural life-style)? We consider a three-component reaction–diffusion system proposed by Aoki, Shida and Shigesada, which describes the interactions among the original farmers, the converted farmers, and the hunter-gatherers. In order to resolve these two questions, we discuss traveling wave solutions which give the information of the expanding velocity of farmer populations. The main result is that two types of traveling wave solutions exist, depending on the growth rate of the original farmer population and the conversion rate of the hunter-gatherer population to the converted farmer population. The profiles of traveling wave solutions indicate that the expansion of farmer populations is determined by the growth rate of the original farmer and the (maximal) carrying capacity of the converted farmer, and the fate of hunter-gatherers is determined by the growth rate of the hunter-gatherer and the conversion rate of the hunter-gatherer to the converted farmer. Thus, our results provide a partial answer to the above two questions.
<br>Traveling wave solutions to di ffusive Holling-Tanner predator-prey models
https://ah.nccu.edu.tw/handle/140.119/137564
title: Traveling wave solutions to di ffusive Holling-Tanner predator-prey models abstract: In this paper, we first establish the existence of semi-traveling wave solutions to a diffusive generalized Holling-Tanner predator-prey model in which the functional response may depend on both the predator and prey populations. Then, by constructing the Lyapunov function, we apply the obtained result to show the existence of traveling wave solutions to the diffusive Holling-Tanner predator-prey models with various functional responses, including the Lotka-Volterra type functional response, the Holling type Ⅱ functional response and the Beddington-DeAngelis functional response.
<br>On the topological entropy of subshifts of finite type on free semigroups
https://ah.nccu.edu.tw/handle/140.119/137563
title: On the topological entropy of subshifts of finite type on free semigroups abstract: In this paper, we provide an effective method to compute the topological entropies of GG-subshifts of finite type (GG-SFTs) with G=FdG=Fd and SdSd, the free group and free semigroup with dd generators respectively. We develop the entropy formula by analyzing the corresponding systems of nonlinear recursive equations (SNREs). Four types of SNREs of S2S2-SFTs, namely the types EE, DD, CC and OO, are introduced, and we could compute their entropies explicitly. This enables us to give the complete characterization of S2S2-SFTs on two symbols. That is, the set of entropies of S2S2-SFTs on two symbols is equal to E∪D∪C∪OE∪D∪C∪O. The methods developed in SdSd-SFTs will also apply to the study of the entropy theory of FdFd-SFTs. The entropy formulae of SdSd-, FdFd-golden mean shifts and kk-colored chessboards are also presented herein.
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