5,552 research outputs found
Dirichlet problems for harmonic functions in half spaces
In our paper, we prove that if the positive part u+(x) of a harmonic function u(x) in a half space satisfies the condition of slow growth, then its negative part u−(x) can also be dominated by a similar growth condition. Moreover, we give an integral representation of the function u(x). Further, a solution of the Dirichlet problem in the half space for a rapidly growing continuous boundary function is constructed by using the generalized Poisson integral with this boundary function.Доведено, що у випадку, коли додатна частина u+(x)гармонічної функції u(x) у напiвпросторi задовольняє умову повільного зростання, її від'ємна частина u−(x) також може бути домінована подібною умовою зростання. Крім того, наведено інтегральне зображення для функції u(x). Більш того, розв'язок задачі Діріхле в напівпросторі для швидко зростаючої неперервної граничної функції побудовано за допомогою узагальненого інтеграла Пуассона з цією граничною функцією
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Network Topologies That Can Achieve Dual Function of Adaptation and Noise Attenuation.
Many signaling systems execute adaptation under circumstances that require noise attenuation. Here, we identify an intrinsic trade-off existing between sensitivity and noise attenuation in the three-node networks. We demonstrate that although fine-tuning timescales in three-node adaptive networks can partially mediate this trade-off in this context, it prolongs adaptation time and imposes unrealistic parameter constraints. By contrast, four-node networks can effectively decouple adaptation and noise attenuation to achieve dual function without a trade-off, provided that these functions are executed sequentially. We illustrate ideas in seven biological examples, including Dictyostelium discoideum chemotaxis and the p53 signaling network and find that adaptive networks are often associated with a noise attenuation module. Our approach may be applicable to finding network design principles for other dual and multiple functions
Noise control and utility: From regulatory network to spatial patterning
Stochasticity (or noise) at cellular and molecular levels has been observed
extensively as a universal feature for living systems. However, how living
systems deal with noise while performing desirable biological functions remains
a major mystery. Regulatory network configurations, such as their topology and
timescale, are shown to be critical in attenuating noise, and noise is also
found to facilitate cell fate decision. Here we review major recent findings on
noise attenuation through regulatory control, the benefit of noise via
noise-induced cellular plasticity during developmental patterning, and
summarize key principles underlying noise control
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