Figure 1.
Nochka-weight-diagram.
Citation: Mark P. McHenry, P. V. Brady, M. M. Hightower. PV-Li-ion-micropump membrane systems for portable personal desalination[J]. AIMS Energy, 2016, 4(3): 444-460. doi: 10.3934/energy.2016.3.444
| [1] | Hong Yan Xu, Yu Xian Chen, Jie Liu, Zhao Jun Wu . A fundamental theorem for algebroid function in $ k $-punctured complex plane. AIMS Mathematics, 2021, 6(5): 5148-5164. doi: 10.3934/math.2021305 |
| [2] | Yige Zhao, Yibing Sun, Zhi Liu, Yilin Wang . Solvability for boundary value problems of nonlinear fractional differential equations with mixed perturbations of the second type. AIMS Mathematics, 2020, 5(1): 557-567. doi: 10.3934/math.2020037 |
| [3] | Tongzhu Li, Ruiyang Lin . Classification of Möbius homogeneous curves in $ \mathbb{R}^4 $. AIMS Mathematics, 2024, 9(8): 23027-23046. doi: 10.3934/math.20241119 |
| [4] | Badriah Alamri . Fixed point theory in elliptic-valued metric spaces: applications to Fredholm integral equations and climate change analysis. AIMS Mathematics, 2025, 10(6): 14222-14247. doi: 10.3934/math.2025641 |
| [5] | Haiming Liu, Jiajing Miao . Geometric invariants and focal surfaces of spacelike curves in de Sitter space from a caustic viewpoint. AIMS Mathematics, 2021, 6(4): 3177-3204. doi: 10.3934/math.2021192 |
| [6] | Süleyman Şenyurt, Filiz Ertem Kaya, Davut Canlı . Pedal curves obtained from Frenet vector of a space curve and Smarandache curves belonging to these curves. AIMS Mathematics, 2024, 9(8): 20136-20162. doi: 10.3934/math.2024981 |
| [7] | Yaqiong Liu, Yunting Li, Qiuping Liao, Yunhui Yi . Classification of nonnegative solutions to fractional Schrödinger-Hatree-Maxwell type system. AIMS Mathematics, 2021, 6(12): 13665-13688. doi: 10.3934/math.2021794 |
| [8] | Yan Sun, Xiao-lan Liu, Jia Deng, Mi Zhou . Some fixed point results for $ \alpha $-admissible extended $ \mathcal{Z} $-contraction mappings in extended rectangular $ b $-metric spaces. AIMS Mathematics, 2022, 7(3): 3701-3718. doi: 10.3934/math.2022205 |
| [9] | Rafik Aguech . On the central limit theorem for the elephant random walk with gradually increasing memory and random step size. AIMS Mathematics, 2024, 9(7): 17784-17794. doi: 10.3934/math.2024865 |
| [10] | Talat Körpinar, Yasin Ünlütürk . An approach to energy and elastic for curves with extended Darboux frame in Minkowski space. AIMS Mathematics, 2020, 5(2): 1025-1034. doi: 10.3934/math.2020071 |
In 1929, relating to the study of value distribution theory for meromorphic functions, R. Nevanlinna [3] conjectured that the second main theorem for meromorphic functions is still valid if one replaces the fixed points by meromorphic functions of slow growth. This conjecture was solved by Osgood [4], Steinmetz [8], and Yamanoi [10] with truncation one. In 1991, Ru and Stoll [5] established the second main theorem for linearly nondegenerate holomorphic curves and moving hyperplanes in subgeneral position. We recall Ru and Stoll's result (for notations, see the following review of background materials).
Theorem 1.1 (Ru and Stoll [5]). Let $ f: \mathbb{C}\rightarrow \mathbb{P}^N (\mathbb{C}) $ be a holomorphic map, and let $ H_j, 1\leq j\leq q, $ be the moving hyperplanes in $ \mathbb{P}^N (\mathbb{C}) $ which are given by $ H_j = \{X = [X_0:\cdots:X_n] \ | \ a_{j0}X_0+\cdots+a_{jn}X_n = 0\}, $ where $ a_{j0}, \dots, a_{jn} $ are entire functions without common zeros. Let $ {\mathcal K}_{{\mathcal H}} $ be the smallest field that contains $ \mathbb{C} $ and all $ a_{j\mu}\over a_{j\nu} $ with $ a_{j\nu} \not\equiv 0. $ Suppose that $ {\mathcal H}: = \{H_1, ..., H_q\} $ is a family of slowly moving hyperplanes with respect to $ f $ located in $ m $-subgeneral position. Assume that $ f $ is linearly nondegenerate over $ {\mathcal K}_{{\mathcal H}} $. Then, for any $ \epsilon > 0 $,
| $ \sum\limits_{j = 1}^q m_f( r, H_j) \leq_{exc} (2m-N+1 + \epsilon)T_f( r ), $ |
where "$ \leq_{exc} $" means that the above inequality holds for all $ r $ outside a set with finite Lebesgue measure.
Before stating our main result, we recall some basic definitions for moving targets. Let $ f: \mathbb C \to \mathbb P^N (\mathbb C) $ be a holomorphic map. Denote by $ {\mathbf f} = (f_0, ..., f_N). $ $ {\mathbf f} $ is called a reduced representation of $ f $ if $ \mathbb{P}(f) = f $ and $ f_0, ..., f_N $ are entire functions without common zeros. Let $ \left\| {{\bf f}(z)} \right\| = \max \left\{ {\left| {f_0(z)} \right|, ..., \left| {f_N(z)} \right|} \right\} $. The characteristic function of $ f $ is defined by
| $ {T_f}( r ) = \int_0^{2\pi } {\log \left\| {{\bf f}( re^{i\theta }}) \right\|} \frac{d\theta } {2\pi }. $ |
We say a meromorphic function $ g $ on $ \mathbb C $ is of slow growth with respect to $ f $ if $ T_g(r) = o(T_f(r)) $. Let $ \mathcal K_f $ be the field of all meromorphic functions on $ \mathbb C $ of slow growth with respect to $ f, $ which is a subfield of meromorphic functions on $ \mathbb C $. For a positive integer $ d $, we set
| $ {{\mathcal I}_d}: = \left\{ I = ( i_0, ..., i_N ) \in {\Bbb Z}_{\geq 0}^{N + 1} \ \big| \ {i_0} + \cdots + {i_N} = d \right\} $ |
and
| $ {n_d} = \# {\mathcal I_d} = \left( {d+NN } \right). $ |
A moving hypersurface $ D $ in $ \mathbb P^N(\mathbb C) $ of degree $ d $ is defined by a homogeneous polynomial $ Q = \sum\nolimits_{I \in {\mathcal I_d}} {{a_I}{{\bf x}^I}} $, where $ a_I, I \in \mathcal I_d $, are holomorphic functions on $ \mathbb C $ without common zeros, and $ {\bf x}^I = {x_0}^{i_0}\cdots{x_N}^{i_N} $. Note that $ D $ can be regarded as a holomorphic map $ a:\Bbb C \to {\Bbb P}^{{n_d} - 1}(\Bbb C) $ with a reduced representation $ ({..., {a_I}(z), ... })_{I \in \mathcal I_d} $. We call $ D $ a slowly moving hypersurface with respect to $ f $ if $ T_a(r) = o(T_f(r)) $. The proximity function of $ f $ with respect to the moving hypersurface $ D $ with defining homogeneous polynomial $ Q $ is defined by
| $ m_f(r, D) = \int_0^{2\pi } \lambda _{D( re^{i\theta })}({\bf f}( re^{i\theta } ))\frac{d\theta }{2\pi }, $ |
where $ \lambda _{D(z)}({\bf f}(z)) = \log \frac{{{\left\| {{\bf f}(z)} \right\|}^d}\left\| {Q(z)} \right\|} {\left\| {Q({\bf f})(z)} \right\|} $ is the Weil function associated to $ D $ composites with $ f $ and $ \left\| {Q(z)} \right\| = \mathop {\max }\limits_{I \in \mathcal I_d} \left\{ {\left| {a_I(z)} \right|} \right\} $. If $ D $ is a slowly moving hypersurface with respect to $ f $ of degree $ d $, we have
| $ m_f(r, D) \leq dT_f(r)+o(T_f(r)) $ |
by the first main theorem for moving targets.
Definition 1.2. Under the above notations, we say that $ f $ is linearly nondegenerate over $ \mathcal K_f $ if there is no nonzero linear form $ L \in \mathcal K_f \left[x_0, ..., x_N \right] $ such that $ L(f_0, ..., f_N) \equiv 0 $, and $ f $ is algebraically nondegenerate over $ \mathcal K_f $ if there is no nonzero homogeneous polynomial $ Q \in \mathcal K_f[x_0, ..., x_N] $ such that $ Q(f_0, ..., f_N) \equiv 0 $. If $ f $ is not algebraically nondegenerate over $ \mathcal K_f, $ we say that $ f $ is degenerate over $ \mathcal K_f. $
Remark 1.3. In this paper, we only consider those moving hypersurfaces $ D $ with defining function $ Q $ such that $ Q(f_0, ..., f_N) \not\equiv 0 $.
We say that the moving hypersurfaces $ D_1, \dots, D_q $ are in $ m $-subgeneral position if there exists $ z \in \mathbb C $ such that $ D_1(z), \dots, D_q(z) $ are in $ m $-subgeneral position (as fixed hypersurfaces), i.e., any $ m+1 $ of $ D_1(z), ..., D_q(z) $ do not meet at one point. Actually, if the condition is satisfied for one point $ z \in \mathbb C $, it is also satisfied for all $ z \in \mathbb C $ except for a discrete set.
In 2021, Heier and Levin generalized Schmidt's subspace theorem to closed subschemes in general position by using the concept of Seshadri constant, as shown in [1]. Recently, they extended this result to arbitrary closed subschemes without any assumption by using the notion of distributive constants and weights assigned to subvarieties [2]. As a corollary, they obtained a second main theorem for hypersurfaces in $ m $-subgeneral position in $ \Bbb P^n(\Bbb C) $, establishing an inequality with factor $ 3\over 2 $.
Main Theorem (Heier and Levin [2]). Let $ f : \Bbb C \rightarrow \Bbb P^N(\Bbb C) $ be a holomorphic map, and let $ D_1, \dots, D_q $ be hypersurfaces in $ \Bbb P^N(\Bbb C) $ of degree $ d_1, \dots, d_q, $ respectively. Assume that $ D_1, \dots, D_q $ are located in $ m $-subgeneral position. Then, for any $ \epsilon > 0 $,
| $ q∑j=11djmf(r,Dj)≤exc(32(2m−N+1)+ϵ)Tf(r). $ |
Here "$ \leq_{exc} $" means that the above inequality holds for all $ r $ outside a set with finite Lebesgue measure.
A key point of their proof is the use of the last line segment of the Nochka diagram, where they proved that the slope of this line segment has a lower bound depending solely on $ m $ and $ N $.
In this paper, motivated by Heier and Levin's work, we consider the moving hypersurfaces in $ m $-subgeneral position and prove the following theorem.
Theorem 1.4 (Main Theorem). Let $ f: \Bbb C\rightarrow \Bbb P^N (\Bbb C) $ be a holomorphic map, and let $ D_1, \ldots, D_q $ be a family of slowly moving hypersurfaces with respect to $ f $ of degree $ d_1, \dots, d_q $, respectively. Assume that $ f $ is algebraically nondegenerate over $ \mathcal K_f $ and $ D_1, \dots, D_q $ are located in $ m $-subgeneral position. Then, for any $ \epsilon > 0 $,
| $ \sum\limits_{j = 1}^q {1\over d_j} m_f( r, D_j) \leq_{exc} {3\over 2}(2m-N+1 + \epsilon)T_f( r ). $ |
Here "$ \leq_{exc} $" means that the above inequality holds for all $ r $ outside a set with finite Lebesgue measure.
Indeed, we prove a more general case when $ f $ is degenerate over $ \mathcal K_f $. To do so, we introduce the notion of "universal fields". Let $ k $ be a field. A universal field $ \Omega_k $ of $ k $ is a field extension of $ k $ that is algebraically closed and has infinite transcendence degree over $ k $. A useful fact of $ \Omega_k $ is that any field extension obtained by adjoining finitely many field elements to $ k $ can be isomorphically imbedded in $ \Omega_k, $ which fixes the base field $ k. $
In this paper, we take $ k = \mathcal K_f $ and fix a universal field $ \Omega $ over $ k = \mathcal K_f $. Let $ f = [f_0:f_1:\cdots:f_N] $ be a reduced representation of $ f $. We can regard each $ f_i $, $ 0 \leq i \leq N $, as an element in $ \Omega $. Hence $ f $ can be seen as a set of homogeneous coordinates of some point $ P $ in $ \mathbb P ^N(\Omega) $. Equip $ \mathbb P ^N(\Omega) $ with the natural Zariski topology.
Definition 1.5. Under the above assumptions, we define the closure of $ P $ in $ \mathbb P ^N(\Omega) $ over $ \mathcal K _f, $ denoted by $ V_f, $ by
| $ Vf:=⋂h∈Kf[x0,…,xN],h(f)≡0{[X0:⋯:XN]∈PN(Ω) | h(X0,⋯,XN)=0}⊆PN(Ω). $ | (1.1) |
Note that $ f $ is algebraically nondegenerate over $ \mathcal K_f, $ which is equivalent to $ V_f = \mathbb P ^N(\Omega) $. We also note that every moving hypersurface $ D $ with defining function $ Q\in \mathcal{K}_f[x_0, \dots, x_N] $ can be seen as a hypersurface determined by $ Q $ in $ \mathbb{P}^N(\Omega). $
Let $ V \subset {\Bbb P}^N(\Omega) $ be an algebraic subvariety defined by homogeneous polynomials $ h_1, \dots, h_s \in \mathcal{K}_f[x_0, \dots, x_N]. $ Let $ z $ be a point in $ \Bbb C $ such that all coefficients of $ h_1, \dots, h_s $ are holomorphic at $ z. $ We denote by $ V(z)\subset \Bbb P^N(\Bbb C) $ the algebraic subvariety of $ \Bbb P^N(\Bbb C) $ defined by $ h_1(z), \dots, h_s(z). $ Here if $ h_j = \sum_{I \in {\mathcal I_d}} {{a_I}{{\bf x}^I}}, $ we denote $ h_j(z) $ by $ h_j(z)(x_0, \dots, x_N) = \sum\nolimits_{I \in {\mathcal I_d}} {{a_I(z)}{{\bf x}^I}}\in \Bbb C[x_0, \dots, x_N]. $ We recall Lemma 3.3 in [11].
Lemma 1.6 (Lemma 3.3, [11]). $ \dim V(z) = \dim V $ and $ \deg V(z) = \deg V $ for all $ z\in \Bbb C $ except a discrete subset.
Definition 1.7. Let $ V $ be an algebraic subvariety of $ {\Bbb P}^N(\Omega) $. We say that $ V $ is defined over $ \mathcal{K}_f $ if $ V $ is an algebraic subvariety defined by some homogeneous polynomials in $ \mathcal{K}_f[x_0, \dots, x_N]. $
Let $ V \subset {\Bbb P}^N(\Omega) $ be an algebraic subvariety defined over $ \mathcal{K}_f $ and $ D_1, ..., D_q $ be $ q $ hypersurfaces in $ {\Bbb P}^N(\Omega) $ defined over $ \mathcal{K}_f $. We say that $ D_1, ..., D_q $ are in $ m $-subgeneral position on $ V $ if for any $ J \subset \{1, \cdots, q\} $ with $ {\#}J \leq m+1 $,
| $ \dim{ \cap _{j \in J}}{D_j} \cap V \leq m - {\#}J. $ |
When $ m = n $, we say $ D_1, \cdots, D_q $ are in general position on $ V $. Note that $ \dim{ \cap _{j \in J}}{D_j}(z) \cap V(z) = \dim{ \cap _{j \in J}}{D_j} \cap V $ for all $ z \in \mathbb C $ excluding a discrete subset.
Remark 1.8. By Lemma 1.6, the definition of $ m $-subgeneral position above implies the definition of $ m $-subgeneral position below Remark 1.3.
We prove the following general result.
Theorem 1.9. Let $ f $ be a holomorphic map of $ {\Bbb C} $ into $ {\Bbb P}^N ({\Bbb C}) $. Let $ \mathcal D = \{D_1, \ldots, D_q\} $ be a family of slowly moving hypersurfaces in $ \mathbb P ^N(\mathbb C) $ with respect to $ f $ with $ \deg D_j = d_j (1\leq j \leq q $). Let $ V_f\subset {\Bbb P}^N(\Omega) $ be given as in (1.1). Assume that $ D_1, \dots, D_q $ are in $ m $-subgeneral position on $ V_f $ and $ \dim V_f = n $. Assume that the following Bezout property holds on $ V_f $ for intersections among the divisors: If $ I, J \subset \{1, ..., q\}, $ then
| $ {\mbox{codim}}_{V_f} D_{I\cup J} = {\mbox{codim}}_{V_f}(D_I\cap D_J)\leq {\mbox{codim}}_{V_f}D_I+ {\mbox{codim}}_{V_f}D_J, $ |
where for every subvariety $ Z $ of $ \mathbb{P}^N(\Omega), $ $ {\mbox{codim}}_{V_f}Z $ is given by $ {\mbox{codim}}_{V_f}Z: = {\mbox{dim}}V_f - {\mbox{dim}}V_f\cap Z. $ Then
| $ \sum\limits_{j = 1}^q {1\over d_j} m_f(r, D_j)\leq_{exc} {3\over 2}(2m-n+1+\epsilon)T_f(r). $ |
Remark 1.10. Recall that we only consider those moving hypersurfaces $ D $ with defining function $ Q \in \mathcal K_f [x_0, ..., x_N] $ such that $ Q(f) \not\equiv 0 $. So we have $ V_f \not\subset D_j $ for every $ 1 \leq j \leq q $.
It is known that the Bezout property holds on projective spaces. Therefore, Theorem 1.4 is the special case of Theorem 1.9 when $ V_f = {\Bbb P}^N(\Omega) $. Therefore, the rest of the paper is devoted to proving Theorem 1.9.
In 2022, Quang [6] introduced the notion of distributive constant $ \Delta $ as follows:
Definition 2.1. Let $ f: \mathbb C \to \mathbb P^N (\mathbb C) $ be a holomorphic curve. Let $ D_1, \cdots, D_q $ be $ q $ hypersurfaces in $ {\Bbb P}^N(\Omega) $. Let $ V_f $ be given as in (1.1). We define the distributive constant for $ D_1, ..., D_q $ with respect to $ f $ by
| $ \Delta: = \max\limits_{\Gamma\subset \{1, ... , q\}}{\#\Gamma\over \mbox{codim}_{V_f} (\bigcap\limits_{j\in \Gamma}D_j)}. $ |
We remark that Quang's original definition (see Definition 3.3 in [6]) is different from Definition 2.1. But by Lemma 3.3 in [11], we can see that Definition 2.1 is equivalent to Definition 3.3 in [6]. We rephrase the definition, according to Heier–Levin [2], as follows:
Definition 2.2. With the assumptions and notations in Definition 2.1, for a closed subset $ W $ of $ V_f $ (with respect to the Zariski topology on $ \Bbb P^N(\Omega) $), let
| $ \alpha(W) = \#\{j\; |\; W\subset \mbox{Supp}D_j\}. $ |
We define
| $ \Delta: = \max\limits_{\emptyset \subsetneq W\subsetneq V_f}{\alpha(W)\over \mbox{codim}_{V_f} W}. $ |
We show that the above two definitions are equivalent. Suppose that $ \tilde{W} $ is a subvariety of $ V_f $ such that $ {\alpha(W)\over \mbox{codim}_{V_f} W} $ attains the maximum at $ W = \tilde{W}. $ Reordering if necessary, we assume that $ \tilde{W} \subset D_j $ for $ j = 1, \dots, \alpha(\tilde{W}). $ Let $ W' = \cap_{j = 1}^{\alpha(\tilde{W})}D_j. $ Then, clearly, $ \tilde{W} \subset W' $ and hence $ \mbox{codim}_{V_f} \tilde{W} \geq \mbox{codim}_{V_f} W'. $ On the other hand, $ \tilde{W} \nsubseteq D_j $ for all $ j > \alpha(\tilde{W}) $ implies that $ W' \nsubseteq D_j $ for all $ j > \alpha(\tilde{W}) $. So $ \alpha(\tilde{W}) = \alpha(W'). $ Thus we have
| $ {\alpha(W')\over \mbox{codim}_{V_f} W'} \geq {\alpha(\tilde{W})\over \mbox{codim}_{V_f} \tilde{W}}. $ |
By our assumption for $ \tilde{W} $, we obtain
| $ {\alpha(W')\over \mbox{codim}_{V_f} W'} = {\alpha(\tilde{W})\over \mbox{codim}_{V_f} \tilde{W}}. $ |
This means that, in Definition 2.2, we only need to consider those $ W $ that are the intersections of some $ D_i $'s, and our claim follows from this observation. In the following, when we deal with $ W $, we always assume that $ W $ is the intersection of some $ D_i $'s.
S. D. Quang obtained the following result.
Theorem 2.3 (S. D. Quang [6], Lei Shi, Qiming Yan, and Guangsheng Yu [7]). Let $ f $ be a holomorphic map of $ {\Bbb C} $ into $ {\Bbb P}^N ({\Bbb C}) $. Let $ \{D_j\}_{j = 1}^q $ be a family of slowly moving hypersurfaces in $ {\Bbb P}^N({\Bbb C}) $ with $ \deg D_j = d_j (1\leq j \leq q $). Let $ V_f \subset \mathbb P^N(\Omega) $ be given as in (1.1). Assume that $ \dim V_f = n $. Then, for any $ \epsilon > 0 $,
| $ \sum\limits_{j = 1}^q {1\over d_j} m_f(r, D_j)\leq_{exc} \left((n+1)\max\limits_{\emptyset \subsetneq W\subsetneq V_f}{\alpha(W)\over {\mbox{codim}}_{V_f} W}+\epsilon\right)T_f(r). $ |
We derive the following corollary of Theorem 2.3.
Corollary 2.4. We adopt the assumptions in Theorem 2.3. Let $ W_0 $ be a closed subset of $ V_f\subset {\Bbb P}^N(\Omega) $. Then, for any $ \epsilon > 0, $ we have
| $ \sum\limits_{j = 1}^q {1\over d_j} m_f(r, D_j)\leq_{exc} \left(\alpha(W_0)+(n+1)\max\limits_{\phi \subsetneq W \subsetneq {V_f}} {\alpha(W)-\alpha(W\cup W_0)\over {\mbox{codim}}_{V_f} W}+\epsilon\right)T_{f}(r). $ |
Proof. Without loss of generality, we suppose that $ W_0 \subset \mbox{Supp} D_j $ for $ j = q-\alpha(W_0)+1, \dots, q $. Let $ q' = q-\alpha(W_0) $. Let
| $ \alpha'(W) = \#\{i\leq q'\; |\; W\subset \mbox{Supp}D_i\}. $ |
Note that $ \alpha'(W) = \alpha(W)-\alpha(W \cup W_0) $. Then by the first main theorem for moving targets and $ \alpha(W_0) = q-q' $,
| $ \sum\limits_{i = q'+1}^q {1\over d_i} m_f(r, D_i)\leq (\alpha(W_0)+\epsilon)T_{f}(r). $ |
Thus, Theorem 2.3 implies that
| $ q∑j=11djmf(r,Dj)=q′∑i=11dimf(r,Di)+q∑i=q′+11dimf(r,Di)≤exc((n+1)max∅⊊W⊊Vfα′(W)codimVfW+ϵ)Tf(r)+(α(W0)+ϵ)Tf(r), $ |
which is we desired.
Proof of Theorem 1.9. We divide the proof into two cases. The first case is that for every algebraic subvariety $ W \subset V_f\subset {\Bbb P}^N(\Omega) $ with $ W\not = \emptyset $, we have $ \mbox{codim}_{V_f} W \ge {n+1\over 2m-n+1}\alpha(W) $. Then, by Definition 2.2, we have $ \Delta \leq {2m-n+1 \over n+1} $. So the result follows easily from Theorem 2.3.
Otherwise, we take a subvariety $ W_0 \subset V $ such that the quantity
| $ {n+1-\mbox{codim}_{V_f} W\over 2m-n+1-\alpha(W)} $ |
is maximized at $ W = W_0 $. We assume that $ W_0 $ is an intersection $ D_I $ for some $ I \subset \{1, \cdots, q\} $. Let
| $ \sigma: = {n+1-\mbox{codim}_{V_f} W_0\over 2m-n+1-\alpha(W_0)}. $ |
Note that $ \sigma $ is the slope of the straight line passing through $ (2m-n+1, n+1) $ and $ (\alpha(W_0), \mbox{codim}_{V_f}W_0) $.
Take arbitrary $ \emptyset \subsetneq W \subsetneq V_f $. By Corollary 2.4, it suffices to show that
| $ \alpha(W_0)+(n+1){\alpha(W)-\alpha(W\cup W_0)\over \mbox{codim}_{V_f} W}\leq {3\over 2}(2m-n+1). $ |
Assume that $ W = D_J $ for some nonempty $ J \subset \{1, ..., q\} $ (the case $ \alpha(W) = 0, J = \emptyset $, follows from $ m $-subgeneral position). From the claim on page 19 of Heier–Levin [2] (apply the same argument in [2]), we have
| $ {\alpha(W_0)-\alpha(W\cup W_0)\over \mbox{codim}_{V_f} W}\leq {1\over \sigma}. $ |
Hence
| $ α(W0)+(n+1)α(W)−α(W∪W0)codimVfW≤α(W0)+n+1σ. $ | (3.1) |
Finally, consider Vojta's Nochka-weight-diagram [9] (see Figure 1).
We note that from our assumption that $ \mathrm {codim}_{V_f} W_0 < {n+1 \over 2m-n+1} \alpha (W_0) $, $ P = (\alpha(W_0), \mbox{codim}_{V_f} W_0) $ lies below the line $ y = {n+1\over 2m-n+1} x $. From the $ m $-subgeneral position, it also lies to the left of the line $ y = x+n-m $. Therefore, $ P $ must lie below and to the left of the intersection point $ Q = \left({2m-n+1\over 2}, {n+1\over 2}\right) $ of the above two straight lines. Thus, we have
| $ α(W0)<2m−n+12,codimVfW0<n+12. $ | (3.2) |
Since $ \sigma > {n+1\over 2m-n+1} $ (see Figure 1), by using (3.2), we obtain
| $ α(W0)+n+1σ<α(W0)+2m−n+1<2m−n+12+2m−n+1=32(2m−n+1). $ |
Combing above with (3.1), we obtain
| $ \alpha(W_0)+(n+1){\alpha(W)-\alpha(W\cup W_0)\over \mbox{codim}_{V_f} W} \leq {3\over 2}(2m-n+1). $ |
The theorem thus follows from Corollary 2.4.
It is a longstanding problem in Nevanlinna theory: we expect the second main theorem, under the setting of hypersurfaces in $ {\mathbb P}^N(\mathbb C) $ located in $ m $-subgeneral position, has the upper bound of $ (2m-N+1+\epsilon)T_f(r). $ This problem can also be considered in the context of moving hypersurfaces in $ m $-subgeneral position. Heier and Levin used an estimate on the slope of the last line segment of the Nochka diagram, along with the concept of distributive constants, to obtain a good coefficient $ {3\over 2}(2m-N+1+\epsilon) $ in the case of fixed hypersurfaces. In this paper, building upon the work of Heier and Levin and utilizing the concept of universal fields, we establish a general inequality with the same coefficient $ {3\over 2}(2m-N+1+\epsilon) $ in the case of moving hypersurfaces. From these theorems, it is seen that a more precise estimation of the slopes of the latter line segments in the Nochaka diagram could lead to a more precise upper bound.
Qili Cai: Conceptualization, formal analysis, investigation, visualization, writing – original draft, writing – review and editing; Chinjui Yang: Conceptualization, formal analysis, methodology, validation, writing – original draft, writing – review and editing. All authors have read and approved the final version of the manuscript for publication.
The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.
The authors declare no conflicts of interest.
| [1] |
Belessiotis V, Delyannis E (2001) Water shortage and renewable energies (RE) desalination - possible technological applications. Desalination 139: 133–138. doi: 10.1016/S0011-9164(01)00302-2
|
| [2] |
Banat F, Jwaied N (2008) Economic evaluation of desalination by small-scale autonomous solar-powered membrane distillation units. Desalination 220: 566–573. doi: 10.1016/j.desal.2007.01.057
|
| [3] | El-Nasher AM (2001) The economic feasibility of small solar MED seawater desalination plants for remote arid areas. Desalination 134: 173–186. |
| [4] | Al-Karaghouli A, Renne D, Kazmerski LL (2009) Solar and wind opportunities for water desalination in the Arab regions. Renew Sust Energ Rev 13: 2397–2407. |
| [5] |
Al-Karaghouli A, Renne D, Kazmerski LL (2010) Technical and economic assessment of photovoltaic-driven desalination systems. Renew Energ 35: 323–328. doi: 10.1016/j.renene.2009.05.018
|
| [6] |
Gude VG, Nirmalakhandan N, Deng S (2010) Renewable and sustainable approaches for desalination. Renew Sust Energ Rev 14: 2641–2654. doi: 10.1016/j.rser.2010.06.008
|
| [7] |
Chaibi MT (2000) An overview of solar desalination for domestic and agriculture water needs in remote arid areas. Desalination 127: 119–133. doi: 10.1016/S0011-9164(99)00197-6
|
| [8] |
Risbey J, Kandlikar M, Dowlatabadi H, et al. (1999) Scale, context, and decision making in agricultural adaptation to climate variability and change. Mitig Adapt Strat Gl 4: 137–165. doi: 10.1023/A:1009636607038
|
| [9] |
Soric A, Cesaro R, Perez P, et al. (2012) Eausmose project desalination by reverse osmosis and batteryless solar energy: design for a 1 m3 per day delivery. Desalination 301: 67–74. doi: 10.1016/j.desal.2012.06.013
|
| [10] |
De Munari A, Capao DPS, Richards BS, et al. (2009) Application of solar-powered desalination in a remote town in South Australia. Desalination 248: 72–82. doi: 10.1016/j.desal.2008.05.040
|
| [11] |
Banat F, Qiblawey H, Al-Nasser Q (2012) Design and operation of small-scale photovoltaic-driven reverse osmosis (PV-RO) desalination plant for water supply in rural areas. CWEEE 1: 31–36. doi: 10.4236/cweee.2012.13004
|
| [12] |
Banasiak LJ, Schafer AI (2009) Removal of inorganic trace contaminants by electrodialysis in a remote Australian community. Desalination 248: 48–57. doi: 10.1016/j.desal.2008.05.037
|
| [13] | Bennett R (2013) System and method for water purification and desalination. US, Lockheed Martin Corporation, 10. |
| [14] | Lazarov V, Zarkov Z, Kanchev H, et al. (2012) Compensation of power fluctuations in PV systems with supercapacitors. E+E 47: 48–55. |
| [15] |
Glavin ME, Hurley WG (2012) Optimisation of a photovoltaic battery ultracapacitor hybrid energy storage system. Solar Energ 86: 3009–3020. doi: 10.1016/j.solener.2012.07.005
|
| [16] |
McHenry MP (2009) Remote area power supply system technologies in Western Australia: New developments in 30 years of slow progress. Renew Energ 34: 1348–1353. doi: 10.1016/j.renene.2008.09.008
|
| [17] |
McHenry MP (2009) Why are remote Western Australians installing renewable energy technologies in stand-alone power supply systems? Renew Energ 34: 1252–1256. doi: 10.1016/j.renene.2008.10.003
|
| [18] | Architectural Energy Corporation (1991) Maintenance and operation of stand-alone photovoltaic systems. Albuquerque, New Mexico, and Boulder, Colorado, USA: Sandia National Laboratories. |
| [19] | Anand S, Fernandes BG (2010) Optimal voltage level for DC microgrids. 36th Annual Conference on IEEE Industrial Electronics Society (IECON), Glendale, Arizona, USA. |
| [20] |
Lee KP, Arnot TC, Mattia D (2011) A review of reverse osmosis membrane materials for desalination - Development to date and future potential. J Membrane Sci 370: 1–22. doi: 10.1016/j.memsci.2010.12.036
|
| [21] |
Hassan AF, Fath HES (2013) Review and assessment of the newly developed MD for desalination processes. Desalin Water Treat 51: 574–585. doi: 10.1080/19443994.2012.697273
|
| [22] |
Wang EN, Karnik R (2012) Water desalination: Graphene cleans up water. Nat Nanotechnol 7: 552–554. doi: 10.1038/nnano.2012.153
|
| [23] |
Ahmadun F-R, Pendashteh A, Abdullah LC, et al. (2009) Review of technologies for oil and gas produced water treatment. J Hazard Mater 170: 530–551. doi: 10.1016/j.jhazmat.2009.05.044
|
| [24] | Guillén-Burrieza E, Zaragoza G, Miralles-Cuevas S, et al. (2012) Experimental evaluation of two pilot-scale membrane distillation modules used for solar desalination. J Membrane Sci 409–410: 264–275. |
| [25] | Guillén-Burrieza E, Blanco J, Zaragoza G, et al. (2011) Experimental analysis of an air gap membrane distillation solar desalination pilot system. J Membrane Sci 397: 386–396. |
| [26] | Onsekizoglu P (2012) Membrane distillation: principle, advances, limitations and future prospects in food industry. In: Zereshki S, Ed. Distillation - advances from modeling to applications. Rijeka, Croatia, InTech, 233–266. |
| [27] | Zhu C, Li H, Zeng XC, et al. (2013) Ideal desalination through graphyne-4 membrane: nanopores for quantized water transport. Condensed Matter arXiv: 1307.0208 |
| [28] |
Tang Q, Zhou Z, Chen Z (2013) Graphene-related nanomaterials: tuning properties by functionalization. Nanoscale 5: 4541–4583. doi: 10.1039/c3nr33218g
|
| [29] |
Cohen-Tanugi D, Grossman JC (2012) Water Desalination across Nanoporous Graphene. Nano Letters 12: 3602–3608. doi: 10.1021/nl3012853
|
| [30] | Cohen-Tanugi D (2012) Nanoporous graphene as a desalination membane: a computational study. Department of Materials Science and Engineering, Cambridge, Massachusetts, USA, Massachusetts Institute of Technology. |
| [31] |
Ganesh BM, Isloor AM, Ismail AF (2013) Enhanced hydrophilicity and salt rejection study of graphene oxide-polysulfone mixed matrix membrane. Desalination 313: 199–207. doi: 10.1016/j.desal.2012.11.037
|
| [32] |
Hu M, Mi B (2013) Enabling graphene oxide nanosheets as water separation membranes. Environ Sci Technol 47: 3715–3723. doi: 10.1021/es400571g
|
| [33] |
Cranford SW, Buehler MJ (2011) Mechanical properties of graphyne. Carbon 49: 4111–4121. doi: 10.1016/j.carbon.2011.05.024
|
| [34] | Zheng JJ, Zhao X, Zhao Y, et al. (2013) Two-dimensional carbon compounds derived from graphyne with chemical properties superior to those of graphene. Sci Rep 3: 1271. |
| [35] | Department of Natural Resources Canada. RETScreen Version 4. (2010) Available from: http: //www.nrcan.gc.ca/energy/software-tools/7465. |
| [36] | Goal Zero (2013) Available from: http: //www.goalzero.com/. |
| [37] | KNF (2013) Diaphragm liquid pump data sheet NF 2.35. Available from: http: //www.knfusa.com/pdfs/nf2-35.pdf . |
| [38] | Amouha MA, Gholam RNB, Behnam H (2011) Nanofiltration efficiency in nitrate removal from groundwater: a semi-industrial case study. International Conference on Environmental Engineering and Applications (ICEEA), Shanghai, China. |
| [39] | The Dow Chemical Company (2013) Dow FilmtecTM NF90 nanofiltration elements for commercial systems. Available from: http: //www.dowwaterandprocess.com/en/products/f/filmtec-nf90_4040. |
| [40] | Xie W, Geise GM, Freeman BD, et al. (2012) Polyamide interfacial composite membranes prepared from m-phenylene diamine, trimesoyl chloride and a new disulfonated diamine. J Membrane Sci 403–404: 152–161. |
Mark P. McHenry, P. V. Brady, M. M. Hightower. PV-Li-ion-micropump membrane systems for portable personal desalination[J]. AIMS Energy, 2016, 4(3): 444-460. doi: 10.3934/energy.2016.3.444
DownLoad:
