Sigmoid generalized complementary equation for wet surface evaporation at the scale with changing advections

<p>Understandings the processes and estimating the amount of wet surface evaporation across various scales are crucial to the evaporation research. The Penman (1948) and Priestley-Taylor (1972) equations are derived for a wet patches and an extensive wet surface respectively, with an obviously different effects of advection. However, the evaporation for a wet surface between these two scales is difficult to estimate because of the changing advections. The sigmoid generalized complementary (SGC) equation, which expresses the ratio of actual evaporation (E) to Penman potential evaporation (E<sub>Pen</sub>) as a function of the proportion of the radiation term (E<sub>rad</sub>) in E<sub>Pen</sub>, is used to model the wet surface evaporation process by setting the symmetric parameter to be infinity, and was validated by data from flux sites over a lake site (CN-MLW) from China, a wetland site (US-WPT) from the United State, and a paddy site (JP-MSE) from Japan. The SGC equation robustly describes the growth of E/E<sub>Pen</sub> upon E<sub>rad</sub>/E<sub>Pen</sub> with upper flatness part over the wet surface with significant changing advection effects, and could account for the variation of the Priestley-Taylor coefficient directly. Thus, the SGC equation outperforms the Priestley-Taylor equation with a constant coefficient for estimating wet surface evaporation at the scale with changing advections.</p>

LIMITED ASSET MARKET PARTICIPATION AND DETERMINACY IN THE OPEN ECONOMY

The perception that inflation targeting (IT) runs a high risk of indeterminacy when a significant share of households are too poor to save is an artifact of the closed economy. In the open economy, the Taylor principle is generally valid for both contemporaneous and forward-looking IT. Active policy in contemporaneous IT guarantees determinacy, eccentric cases aside. In forward-looking IT, the scope for active policy is constrained by an upper bound on the Taylor coefficient. The upper bound is insensitive, however, to the share of poor, nonsaving households. Moreover, it can be increased substantially–to a level that does not bind–through reserve sales/purchases that limit exchange rate volatility.

An efficient algorithm for Padé-type approximation of the frequency response for the Helmholtz problem

We consider the map $\mathcal{S}:\mathbb{C}\to H^1_0(\Omega)=\{v\in H^1(D), v|_{\partial\Omega}=0\}$, which associates a complex value z with the weak solution of the (complex-valued) Helmholtz problem $-\Delta u-zu=f$ over $\Omega$ for some fixed $f\in L^2(\Omega)$. We show that $\mathcal{S}$ is well-defined and meromorphic in $\mathbb{C}\setminus\Lambda$, $\Lambda=\{\lambda_\alpha\}_{\alpha=1}^\infty$ being the (countable, unbounded) set of (real, non-negative) eigenvalues of the Laplace operator (restricted to $H^1_0(\Omega)$). In particular, it holds $\mathcal{S}(z)=\sum_{\alpha=1}^\infty\frac{s_\alpha}{\lambda_\alpha-z}$, where the elements of $\{s_\alpha\}_{\alpha=1}^\infty\subset H^1_0(\Omega)$ are pair-wise orthogonal with respect to the $H^1_0(\Omega)$ inner product. We define a Pad\'e-type approximant of any map as above around $z_0\in\mathbb{C}$: given some integer degrees of the numerator and denominator respectively, $M,N\in\mathbb{N}$, the exact map is approximated by a rational map $\mathcal{S}_{[M/N]}:\mathbb{C}\setminus\Lambda\to H^1_0(\Omega)$. We define such approximant within a Least-Squares framework, through the minimization of a suitable functional based on samples of the target solution map and of its derivatives at $z_0$. In particular, the denominator of the approximant is the minimizer (under some normalization constraints) of the $H^1_0(\Omega)$ norm of a Taylor coefficient of $Q\mathcal{S}$, as Q varies in the space of polynomials with degree $\leq N$. The numerator is then computed by matching as many terms as possible of the Taylor series of $\mathcal{S}$ with those of $\mathcal{S}_{[M/N]}$, analogously to the classical Pad\'e approach. The resulting approximant is shown to converge, as $M+N$ goes to infinity, to the exact map $\mathcal{S}_{[M/N]}$ in the $H^1_0(\Omega)$ norm for values of the parameter sufficiently close to $z_0$ (a sharp bound on the region of convergence is given). Moreover, it is proven that the approximate poles converge exponentially (as M goes to infinity) to the N elements of $\Lambda$ closer to $z_0$.

Boundary Schwarz lemma for holomorphic functions

In this paper, a boundary version of Schwarz lemma is investigated. We take into consideration a function f (z) holomorphic in the unit disc and f (0) = 0 such that ?Rf? < 1 for ?z? < 1, we estimate a modulus of angular derivative of f (z) function at the boundary point b with f (b) = 1, by taking into account their first nonzero two Maclaurin coefficients. Also, we shall give an estimate below ?f'(b)? according to the first nonzero Taylor coefficient of about two zeros, namely z=0 and z0 ? 0. Moreover, two examples for our results are considered.

Matching the Budyko functions with the complementary evaporation relationship: consequences for the drying power of the air and the Priestley–Taylor coefficient

The Budyko functions B1(Φp) are dimensionless relationships relating the ratio E / P (actual evaporation over precipitation) to the aridity index Φp = Ep / P (potential evaporation over precipitation). They are valid at catchment scale with Ep generally defined by Penman's equation. The complementary evaporation (CE) relationship stipulates that a decreasing actual evaporation enhances potential evaporation through the drying power of the air which becomes higher. The Turc–Mezentsev function with its shape parameter λ, chosen as example among various Budyko functions, is matched with the CE relationship, implemented through a generalised form of the advection–aridity model. First, we show that there is a functional dependence between the Budyko curve and the drying power of the air. Then, we examine the case where potential evaporation is calculated by means of a Priestley–Taylor type equation (E0) with a varying coefficient α0. Matching the CE relationship with the Budyko function leads to a new transcendental form of the Budyko function B1′(Φ0) linking E / P to Φ0 = E0 / P. For the two functions B1(Φp) and B1′(Φ0) to be equivalent, the Priestley–Taylor coefficient α0 should have a specified value as a function of the Turc–Mezentsev shape parameter and the aridity index. This functional relationship is specified and analysed.