Citation: Rebecca L. Thomas, Kristin S. Shrader-Frechette. Resolving scientific controversy over smelter risks and neurodegenerative effects of metals[J]. AIMS Environmental Science, 2015, 2(1): 56-86. doi: 10.3934/environsci.2015.1.56
| [1] | Ruiping Yuan, Jiangtao Dou, Juntao Li, Wei Wang, Yingfan Jiang . Multi-robot task allocation in e-commerce RMFS based on deep reinforcement learning. Mathematical Biosciences and Engineering, 2023, 20(2): 1903-1918. doi: 10.3934/mbe.2023087 |
| [2] | Yangjie Sun, Xiaoxi Che, Nan Zhang . 3D human pose detection using nano sensor and multi-agent deep reinforcement learning. Mathematical Biosciences and Engineering, 2023, 20(3): 4970-4987. doi: 10.3934/mbe.2023230 |
| [3] | Jin Zhang, Nan Ma, Zhixuan Wu, Cheng Wang, Yongqiang Yao . Intelligent control of self-driving vehicles based on adaptive sampling supervised actor-critic and human driving experience. Mathematical Biosciences and Engineering, 2024, 21(5): 6077-6096. doi: 10.3934/mbe.2024267 |
| [4] | Siqi Chen, Ran Su . An autonomous agent for negotiation with multiple communication channels using parametrized deep Q-network. Mathematical Biosciences and Engineering, 2022, 19(8): 7933-7951. doi: 10.3934/mbe.2022371 |
| [5] | Shixuan Yao, Xiaochen Liu, Yinghui Zhang, Ze Cui . An approach to solving optimal control problems of nonlinear systems by introducing detail-reward mechanism in deep reinforcement learning. Mathematical Biosciences and Engineering, 2022, 19(9): 9258-9290. doi: 10.3934/mbe.2022430 |
| [6] | Siqi Chen, Yang Yang, Ran Su . Deep reinforcement learning with emergent communication for coalitional negotiation games. Mathematical Biosciences and Engineering, 2022, 19(5): 4592-4609. doi: 10.3934/mbe.2022212 |
| [7] | Jia Mian Tan, Haoran Liao, Wei Liu, Changjun Fan, Jincai Huang, Zhong Liu, Junchi Yan . Hyperparameter optimization: Classics, acceleration, online, multi-objective, and tools. Mathematical Biosciences and Engineering, 2024, 21(6): 6289-6335. doi: 10.3934/mbe.2024275 |
| [8] | Jingxu Xiao, Chaowen Chang, Yingying Ma, Chenli Yang, Lu Yuan . Secure multi-path routing for Internet of Things based on trust evaluation. Mathematical Biosciences and Engineering, 2024, 21(2): 3335-3363. doi: 10.3934/mbe.2024148 |
| [9] | Koji Oshima, Daisuke Yamamoto, Atsuhiro Yumoto, Song-Ju Kim, Yusuke Ito, Mikio Hasegawa . Online machine learning algorithms to optimize performances of complex wireless communication systems. Mathematical Biosciences and Engineering, 2022, 19(2): 2056-2094. doi: 10.3934/mbe.2022097 |
| [10] | Jose Guadalupe Beltran-Hernandez, Jose Ruiz-Pinales, Pedro Lopez-Rodriguez, Jose Luis Lopez-Ramirez, Juan Gabriel Avina-Cervantes . Multi-Stroke handwriting character recognition based on sEMG using convolutional-recurrent neural networks. Mathematical Biosciences and Engineering, 2020, 17(5): 5432-5448. doi: 10.3934/mbe.2020293 |
Let $ {\mathbb C} $ be the complex plane. Denote by $ {\mathbb C}^N $ the $ N $-dimensional complex Euclidean space with the inner product $ \langle z, w\rangle = \sum_{j = 1}^Nz_j\overline{w}_j $; by $ |z|^2 = \langle z, z\rangle $; by $ H({\mathbb C}^N) $ the set of all holomorphic functions on $ {\mathbb C}^N $; and by $ I $ the identity operator on $ {\mathbb C}^N $.
The Fock space $ \mathcal{F}^2({\mathbb C}^N) $ is a Hilbert space of all holomorphic functions $ f\in H({\mathbb C}^N) $ with the inner product
|
$ ⟨f,g⟩=1(2π)N∫CNf(z)¯g(z)e−12|z|2dν(z), $
|
where $ \nu(z) $ denotes Lebesgue measure on $ {\mathbb C}^N $. To simplify notation, we will often use $ \mathcal{F}^2 $ instead of $ \mathcal{F}^2({\mathbb C}^N) $, and we will denote by $ \|f\| $ the corresponding norm of $ f $. The reproducing kernel functions of the Fock space are given by
| $ K_w(z) = e^{\frac{\langle z,w\rangle}{2}},\,\,\,z\in {\mathbb C}^N, $ |
which means that if $ f\in\mathcal{F}^2 $, then $ f(z) = \langle f, K_z\rangle $ for all $ z\in {\mathbb C}^N $. It is easy to see that $ \|K_w\| = e^{|w|^2/4} $. Therefore, the following evaluation holds:
| $ |f(z)|\leq e^{\frac{|z|^2}{4}}\|f\| $ |
for $ f\in\mathcal{F}^2 $ and $ z\in {\mathbb C}^N $. If $ k_w $ is the normalization of $ K_w $, then
| $ k_w(z) = e^{\frac{\langle z,w\rangle}{2}-\frac{|w|^2}{4}},\,\,\,z\in {\mathbb C}^N. $ |
Indeed, $ \mathcal{F}^2 $ is used to describe systems with varying numbers of particles in the states of quantum harmonic oscillators. On the other hand, the reproducing kernels in $ \mathcal{F}^2 $ are used to describe the coherent states in quantum physics. See [17] for more about the Fock space, and see [1,7,11] for the studies of some operators on the Fock space.
For a given holomorphic mapping $ {\varphi}: {\mathbb C}^N\to {\mathbb C}^N $ and $ u\in H({\mathbb C}^N) $, the weighted composition operator, usually denoted by $ W_{u, {\varphi}} $, on or between some subspaces of $ H({\mathbb C}^N) $ is defined by
| $ W_{u,{\varphi}}f(z) = u(z)f({\varphi}(z)). $ |
When $ u = 1 $, it is the composition operator, usually denoted by $ C_{\varphi} $. While $ {\varphi}(z) = z $, it is the multiplication operator, usually denoted by $ M_{u} $.
Forelli in [8] proved that the isometries on Hardy space $ H^p $ defined on the open unit disk (for $ p\neq2 $) are certain weighted composition operators, which can be regarded as the earliest presence of the weighted composition operators. Weighted composition operators have also been used in descriptions of adjoints of composition operators (see [4]). An elementary problem is to provide function-theoretic characterizations for which the symbols $ u $ and $ {\varphi} $ induce a bounded or compact weighted composition operator on various holomorphic function spaces. There have been many studies of the weighted composition operators and composition operators on holomorphic function spaces. For instance, several authors have recently worked on the composition operators and weighted composition operators on Fock space. For the one-variable case, Ueki [13] characterized the boundedness and compactness of weighted composition operators on Fock space. As a further work of [13], Le [10] found the easier criteria for the boundedness and compactness of weighted composition operators. Recently, Bhuia in [2] characterized a class of $ C $-normal weighted composition operators on Fock space.
For the several-variable case, Carswell et al. [3] studied the boundedness and compactness of composition operators. From [3], we see that the one-variable case composition operator $ C_{\varphi} $ is bounded on Fock space if and only if $ {\varphi}(z) = az+b $, where $ |a|\leq 1 $, and if $ |a| = 1 $, then $ b = 0 $. Let $ A: {\mathbb C}^N\to {\mathbb C}^N $ be a linear operator. Zhao [14,15,16] characterized the unitary, invertible, and normal weighted composition operator $ W_{u, {\varphi}} $ on Fock space, when $ {\varphi}(z) = Az+b $ and $ u = k_c $. Interestingly enough, Zhao [15] proved that for $ {\varphi}(z) = Az+b $ and $ u(z) = K_c(z) $, weighted composition operator $ W_{u, {\varphi}} $ is bounded on Fock space if and only if $ \|A\|\leq1 $ and $ \langle A\zeta, b+Ac\rangle = 0 $ whenever $ |A\zeta| = |\zeta| $ for $ \zeta\in {\mathbb C}^N $.
Motivated by the above-mentioned interesting works, for the special symbols $ {\varphi}(z) = Az+b $ and $ u = K_c $, here we study the adjoint, self-adjointness, and hyponormality of weighted composition operators on Fock space. Such properties of the abstract or concrete operators (for example, Toeplitz operators, Hankel operators, and composition operators) have been extensively studied on some other holomorphic function spaces. This paper can be regarded as a continuation of the weighted composition operators on Fock space.
In this section, we characterize the adjoints of weighted composition operators $ W_{u, {\varphi}} $ on Fock space, where $ {\varphi}(z) = Az+b $ and $ u = K_{c} $.
We first have the following result:
Lemma 2.1. Let $ A $, $ B: {\mathbb C}^N\to {\mathbb C}^N $ be linear operators with $ \|A\|\leq1 $ and $ \|B\|\leq1 $, $ {\varphi}(z) = Az+a $, $ \psi(z) = Bz+b $ for $ a, b\in {\mathbb C}^N $, and the operators $ C_{\varphi} $ and $ C_\psi $ be bounded on $ \mathcal{F}^2 $. Then
|
$ C∗φCψ=WKa,BA∗z+b, $
|
where $ A^* $ is the adjoint operator of $ A $.
Proof. From Lemma 2 in [3], it follows that
|
$ C∗φCψ=MKaCA∗zCBz+b=MKaC(Bz+b)∘A∗z=MKaCBA∗z+b=WKa,BA∗z+b, $
|
from which the result follows. The proof is complete.
In Lemma 2.1, we prove that the product of the adjoint of a composition operator and another composition operator is expressed as a weighted composition operator. Next, we will see that in some sense, the converse of Lemma 2.1 is also true. Namely, we will prove that if $ {\varphi}(z) = Az+b $, where $ A: {\mathbb C}^N\to {\mathbb C}^N $ is a linear operator with $ \|A\| < 1 $, and $ u = K_c $, then the operator $ W_{u, {\varphi}} $ on $ \mathcal{F}^2 $ can be written as the product of the adjoint of a composition operator and another composition operator.
Lemma 2.2. Let $ A: {\mathbb C}^N\to {\mathbb C}^N $ be a linear operator with $ \|A\| < 1 $. If $ A $ and $ c $ satisfy the condition $ \langle A^*\zeta, c\rangle = 0 $ whenever $ |A^*\zeta| = |\zeta| $, then there exists a positive integer $ n $ such that the operator $ W_{u, {\varphi}} $ on $ \mathcal{F}^2 $ defined by $ {\varphi}(z) = Az+b $ and $ u(z) = K_c(z) $ is expressed as
|
$ Wu,φ=C∗n+1nA∗z+cCnn+1z+b. $
|
Proof. From Theorem 2 in [3], we see that the operator $ C_{A^*z+c} $ is bounded on $ \mathcal{F}^2 $. Since $ \|A^*\| < 1 $, there exists a large enough positive integer $ n $ such that
|
$ ‖(1+1n)A∗‖≤1. $
|
Also, by Theorem 2 in [3], the operator $ C_{\frac{n+1}{n}A^*z+c} $ is bounded on $ \mathcal{F}^2 $, which implies that the operator $ C^*_{\frac{n+1}{n}A^*z+c} $ is also bounded on $ \mathcal{F}^2 $. Since $ |\frac{n}{n+1}I\zeta| = |\zeta| $ if and only if $ \zeta = 0 $, $ \langle\frac{n}{n+1}I\zeta, b\rangle = 0 $ whenever $ |\frac{n}{n+1}I\zeta| = |\zeta| $. By Theorem 2 in [3], the operator $ C_{\frac{n}{n+1}Iz+b} $ is bounded on $ \mathcal{F}^2 $. Then, it follows from Lemma 2.1 that
|
$ C∗n+1nA∗z+cCnn+1Iz+b=WKc,Az+b. $
|
The proof is complete.
Now, we can obtain the adjoint for some weighted composition operators.
Theorem 2.1. Let $ {\varphi}(z) = Az+b $, $ u(z) = K_c(z) $, and $ A $ and $ c $ satisfy $ \langle A^*\zeta, c\rangle = 0 $ whenever $ |A^*\zeta| = |\zeta| $. Then it holds that
|
$ W∗u,φ=WKb,A∗z+c. $
|
Proof. In Lemma 2.2, we have
|
$ Wu,φ=C∗n+1nA∗z+cCnn+1Iz+b. $
|
(2.1) |
It follows from (2.1) that
|
$ W∗u,φ=C∗nn+1Iz+bCn+1nA∗z+c. $
|
(2.2) |
Therefore, from (2.2) and Lemma 2.1, the desired result follows. The proof is complete.
By using the kernel functions, we can obtain the following result:
Lemma 2.3. Let the operator $ W_{u, {\varphi}} $ be a bounded operator on $ \mathcal{F}^2 $. Then it holds that
|
$ W∗u,φKw=¯u(w)Kφ(w). $
|
Proof. Let $ f $ be an arbitrary function in $ \mathcal{F}^2 $. We see that
|
$ ⟨W∗u,φKw,f⟩=⟨Kw,Wu,φf⟩=¯⟨Wu,φf,Kw⟩=¯u(w)f(φ(w))=¯u(w)⟨Kφ(w),f⟩. $
|
From this, we deduce that $ W^*_{u, {\varphi}}K_w = \overline{u(w)}K_{{\varphi}(w)} $. The proof is complete.
Here, we characterize the self-adjoint weighted composition operators.
Theorem 2.2. Let $ A: {\mathbb C}^N\to {\mathbb C}^N $ be a linear operator, $ b, c\in {\mathbb C}^N $, $ {\varphi}(z) = Az+b $, $ u(z) = K_c(z) $, and the operator $ W_{u, {\varphi}} $ be bounded on $ \mathcal{F}^2 $. Then the operator $ W_{u, {\varphi}} $ is self-adjoint on $ \mathcal{F}^2 $ if and only if $ A: {\mathbb C}^N\to {\mathbb C}^N $ is self-adjoint and $ b = c $.
Proof. In Lemma 2.3, we have
|
$ W∗u,φKw(z)=¯u(w)Kφ(w)=¯Kc(w)e⟨z,φ(w)⟩2=e⟨c,w⟩2e⟨z,Aw+b⟩2. $
|
(2.3) |
On the other hand,
|
$ Wu,φKw(z)=u(z)Kw(φ(z))=e⟨z,c⟩2e⟨Az+b,w⟩2. $
|
(2.4) |
It is clear that operator $ W_{u, {\varphi}} $ is self-adjoint on $ \mathcal{F}^2 $ if and only if
| $ W^*_{u,{\varphi}}K_w = W_{u,{\varphi}}K_w. $ |
From (2.3) and (2.4), it follows that
|
$ e⟨c,w⟩2e⟨z,Aw+b⟩2=e⟨z,c⟩2e⟨Az+b,w⟩2. $
|
(2.5) |
Letting $ z = 0 $ in (2.5), we obtain that $ e^{\frac{\langle c, w\rangle}{2}} = e^{\frac{\langle b, w\rangle}{2}} $ which implies that
|
$ ⟨c,w⟩−⟨b,w⟩=4kπi, $
|
(2.6) |
where $ k\in {\mathbb N} $. Also, letting $ w = 0 $ in (2.6), we see that $ k = 0 $. This shows that $ \langle c, w\rangle-\langle b, w\rangle = 0 $, that is, $ \langle c, w\rangle = \langle b, w\rangle $. From this, we deduce that $ b = c $. Therefore, (2.5) becomes $ e^{\frac{\langle z, Aw\rangle}{2}} = e^{\frac{\langle Az, w\rangle}{2}} $. From this, we obtain that $ \langle z, Aw\rangle = \langle Az, w\rangle $, which implies that $ \langle A^*z, w\rangle = \langle Az, w\rangle $. This shows that $ A = A^* $, that is, $ A: {\mathbb C}^N\to {\mathbb C}^N $ is self-adjoint.
Now, assume that $ A $ is a self-adjoint operator on $ {\mathbb C}^N $ and $ b = c $. A direct calculation shows that (2.5) holds. Then $ W_{u, {\varphi}} $ is a self-adjoint operator on $ \mathcal{F}^2 $. The proof is complete.
In [14], Zhao proved that the operator $ W_{u, {\varphi}} $ on $ \mathcal{F}^2 $ is unitary if and only if there exist an unitary operator $ A: {\mathbb C}^N\to {\mathbb C}^N $, a vector $ b\in {\mathbb C}^N $, and a constant $ {\alpha} $ with $ |{\alpha}| = 1 $ such that $ {\varphi}(z) = Az-b $ and $ u(z) = {\alpha} K_{A^{-1}b}(z) $. Without loss of generality, here we characterize the self-adjoint unitary operator $ W_{u, {\varphi}} $ on $ \mathcal{F}^2 $ for the case $ {\alpha} = 1 $ and obtain the following result from Theorem 2.2.
Corollary 2.1. Let $ A: {\mathbb C}^N\to {\mathbb C}^N $ be a unitary operator and $ b\in {\mathbb C}^N $ such that $ {\varphi}(z) = Az-b $ and $ u(z) = K_{A^{-1}b}(z) $. Then the operator $ W_{u, {\varphi}} $ is self-adjoint on $ \mathcal{F}^2 $ if and only if $ A: {\mathbb C}^N\to {\mathbb C}^N $ is self-adjoint and $ Ab+b = 0 $.
First, we recall the definition of hyponormal operators. An operator $ T $ on a Hilbert space $ H $ is said to be hyponormal if $ \|Ax\|\geq\|A^*x\| $ for all vectors $ x\in H $. $ T $ is called co-hyponormal if $ T^* $ is hyponormal. In 1950, Halmos, in his attempt to solve the invariant subspace problem, extended the notion of normal operators to two new classes, one of which is now known as the hyponormal operator (see [9]). Clearly, every normal operator is hyponormal. From the proof in [6], it follows that $ T $ is hyponormal if and only if there exists a linear operator $ C $ with $ \|C\|\leq1 $ such that $ T^* = CT $. In some sense, this result can help people realize the characterizations of the hyponormality of some operators. For example, Sadraoui in [12] used this result to characterize the hyponormality of composition operators defined by the linear fractional symbols on Hardy space. On the other hand, some scholars studied the hyponormality of composition operators on Hardy space by using the fact that the operator $ C_{{\varphi}} $ on Hardy space is hyponormal if and only if
| $ \|C_{{\varphi}}f\|^2\geq\|C^*_{{\varphi}}f\|^2 $ |
for all $ f $ in Hardy space. For example, Dennis in [5] used the fact to study the hyponormality of composition operators on Hardy space. In particular, this inequality for norms is used when $ f $ is a reproducing kernel function $ K_w $ for any $ w\in {\mathbb C}^N $. Actually, to the best of our knowledge, there are few studies on the hyponormality of weighted composition operators. Here, we consider this property of weighted composition operators on Fock space.
First, we have the following result, which can be proved by using the reproducing kernel functions.
Lemma 3.1. Let $ w\in {\mathbb C}^N $ and the operator $ W_{u, {\varphi}} $ be bounded on $ \mathcal{F}^2 $. Then
|
$ ‖Wu,φKw‖2=W∗u,φWu,φKw(w). $
|
Proof. From the inner product, we have
|
$ ‖Wu,φKw‖2=⟨Wu,φKw,Wu,φKw⟩=⟨W∗u,φWu,φKw,Kw⟩=W∗u,φWu,φKw(w). $
|
The proof is complete.
Theorem 3.1. Let $ A: {\mathbb C}^N\to {\mathbb C}^N $ be a linear operator, $ {\varphi}(z) = Az+b $, $ u = k_c $, and the operator $ W_{u, {\varphi}} $ be bounded on $ \mathcal{F}^2 $. If the operator $ W_{u, {\varphi}} $ is hyponormal on $ \mathcal{F}^2 $, then $ A^*b-b = Ac-c $ and $ |b|\leq|c| $.
Proof. From a direct calculation, we have
|
$ Wu,φKw(z)=u(z)Kw(φ(z))=kc(z)Kw(Az+b)=e⟨z,c⟩2−|c|24e⟨Az+b,w⟩2=e⟨z,A∗w+c⟩+⟨b,w⟩2−|c|24=e⟨b,w⟩2−|c|24KA∗w+c(z). $
|
(3.1) |
From (3.1), it follows that
|
$ W∗u,φWu,φKw(z)=e⟨b,w⟩2−|c|24W∗u,φKA∗w+c(z)=e⟨b,w⟩2−|c|24¯u(A∗w+c)Kφ(A∗w+c)(z)=e⟨b,w⟩2+⟨c,A∗w+c⟩2+⟨z,AA∗w+Ac+b⟩2−|c|22=e⟨b+Ac,w⟩2+⟨z,AA∗w⟩2+⟨z,Ac+b⟩2. $
|
(3.2) |
On the other hand, we also have
|
$ Wu,φW∗u,φKw(z)=¯u(w)Wu,φKφ(w)(z)=¯u(w)u(z)Kφ(w)(φ(z))=e⟨c,w⟩2+⟨z,c⟩2+⟨Az+b,Aw+b⟩2−|c|22=e⟨c+A∗b,w⟩2+|b|22+⟨z,AA∗w⟩2+⟨z,c+A∗b⟩2−|c|22. $
|
(3.3) |
From Lemma 3.1, (3.2), and (3.3), it follows that
|
$ ‖W∗u,φKw‖2=Wu,φW∗u,φKw(w)=e⟨c+A∗b,w⟩2+|b|22+|A∗w|22+⟨w,c+A∗b⟩2−|c|22 $
|
and
|
$ ‖Wu,φKw‖2=W∗u,φWu,φKw(w)=e⟨b+Ac,w⟩2+|A∗w|22+⟨w,Ac+b⟩2. $
|
Then, we have
|
$ ‖W∗u,φKw‖2−‖Wu,φKw‖2=e|A∗w|22(e⟨c+A∗b,w⟩2+|b|22+⟨w,c+A∗b⟩2−|c|22−e⟨b+Ac,w⟩2+⟨w,Ac+b⟩2), $
|
which shows that
| $ \|W^*_{u,{\varphi}}K_w\|^2-\|W_{u,{\varphi}}K_w\|^2\leq0 $ |
for all $ w\in {\mathbb C}^N $ if and only if
|
$ e⟨c+A∗b,w⟩2+|b|22+⟨w,c+A∗b⟩2−|c|22≤e⟨b+Ac,w⟩2+⟨w,Ac+b⟩2. $
|
(3.4) |
It is clear that (3.4) holds if and only if
|
$ ⟨c+A∗b,w⟩+|b|2+⟨w,c+A∗b⟩−|c|2≤⟨b+Ac,w⟩+⟨w,Ac+b⟩. $
|
(3.5) |
From (3.5), we see that (3.4) holds if and only if
|
$ ⟨A∗b−Ac+c−b,w⟩+⟨w,A∗b−Ac+c−b⟩+|b|2−|c|2≤0. $
|
(3.6) |
Therefore, we deduce that (3.4) holds for all $ w\in {\mathbb C}^N $ if and only if $ |b|\leq|c| $ and $ A^*b-b = Ac-c $. The proof is complete.
If $ b = c = 0 $ in Theorem 3.1, then $ W_{u, {\varphi}} $ is reduced into the composition operator $ C_{Az} $. For this case, Theorem 3.1 does not provide any useful information on the operator $ A: {\mathbb C}^N\to {\mathbb C}^N $ when $ C_{Az} $ is hyponormal on $ \mathcal{F}^2 $. However, we have the following result, which completely characterizes the hyponormal composition operators:
Theorem 3.2. Let $ A: {\mathbb C}^N\to {\mathbb C}^N $ be a linear operator such that $ C_{Az} $ is bounded on $ \mathcal{F}^2 $. Then the operator $ C_{Az} $ is hyponormal on $ \mathcal{F}^2 $ if and only if $ A: {\mathbb C}^N\to {\mathbb C}^N $ is co-hyponormal.
Proof. Assume that $ A: {\mathbb C}^N\to {\mathbb C}^N $ is co-hyponormal. Then there exists an operator $ B: {\mathbb C}^N\to {\mathbb C}^N $ with $ \|B\|\leq1 $ such that $ A = BA^* $. We therefore have
| $ C^*_{Az} = C_{A^*z} = C_{AB^*z} = C_{B^*z}C_{Az}. $ |
Next, we want to show that $ \|C_{B^*z}\| = 1 $. By Theorem 4 in [3], we have
|
$ ‖CB∗z‖=e14(|w0|2−|B∗w0|2), $
|
(3.7) |
where $ w_0 $ is any solution to $ (I-BB^*)w = 0 $. From this, we obtain that $ w_0 = BB^*w_0 $, and then
|
$ |B∗w0|2=⟨B∗w0,B∗w0⟩=⟨w0,BB∗w0⟩=⟨w0,w0⟩=|w0|2. $
|
(3.8) |
Thus, by considering (3.7) and (3.8), we see that $ \|C_{B^*z}\| = 1 $. It follows that the operator $ C_{Az} $ is hyponormal on $ \mathcal{F}^2 $.
Now, assume that the operator $ C_{Az} $ is hyponormal on $ \mathcal{F}^2 $. Then there exists a linear operator $ C $ on $ \mathcal{F}^2 $ with $ \|C\|\leq1 $ such that $ C^*_{Az} = CC_{Az} $. By Lemma 2 in [3], we have $ C^*_{Az} = C_{A^*z} $. This shows that $ CC_{Az} $ is a composition operator. This result shows that there exists a holomorphic mapping $ {\varphi}: {\mathbb C}^N\to {\mathbb C}^N $ such that $ C = C_{\varphi} $. So $ A^*z = A({\varphi}(z)) $ for all $ z\in {\mathbb C}^N $, which implies that there exists a linear operator $ B: {\mathbb C}^N\to {\mathbb C}^N $ such that $ {\varphi}(z) = B^*z $, and then $ C = C_{B^*z} $. Therefore, $ A^* = AB^* $, that is, $ A = BA^* $. Since $ \|C\|\leq1 $, this shows that the operator $ C = C_{B^*z} $ is bounded on $ \mathcal{F}^2 $. From Lemma 2.3 in [15], we obtain that $ \|B^*\|\leq1 $, which also shows that $ \|B\|\leq1 $ since $ \|B^*\| = \|B\| $. We prove that $ A: {\mathbb C}^N\to {\mathbb C}^N $ is co-hyponormal. The proof is complete.
Remark 3.1. In the paper, we only obtain a necessary condition for the hyponormality of weighted composition operators on Fock space. We hope that the readers can continuously consider the problem in Fock space.
In this paper, I give a proper description of the adjoint $ W^*_{u, {\varphi}} $ on Fock space for the special symbol functions $ u(z) = K_c(z) $ and $ {\varphi}(z) = Az+b $. However, it is difficult to give a proper description of the general symbols. On the other hand, I consider the hyponormal weighted composition operators on Fock space and completely characterize hyponormal composition operators on this space. I hope that people are interested in the research in this paper.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This study was supported by Sichuan Science and Technology Program (2024NSFSC0416).
The author declares that he has no competing interests.
| [1] | US Congress, House of Representatives: Poison Harvest. Government Printing Office, Washington, D.C, 2012. Available from: http: //www.gpo.gov/fdsys/pkg/CHRG-112hhrg75164/html/CHRG-112hhrg75164.htm. |
| [2] | Kabir E, Sharmila R, Kim K, et al. (2012) Current Status of Trace Metal Pollution in Soils Affected by Industrial Activities. Sci World J 2012: 1-18. |
| [3] |
Grandjean P, Landrigan PJ (2014) Neurobehavioural Effects of Developmental Toxicity. Lancet Neurol 13: 330-338. doi: 10.1016/S1474-4422(13)70278-3
|
| [4] |
Centeno JA, Tseng CH, Van der Voet GB, et al. (2007) Global impacts of geogenic arsenic. Ambio 36: 78-81. doi: 10.1579/0044-7447(2007)36[78:GIOGAA]2.0.CO;2
|
| [5] |
Hu H, Shine J, Wright RO (2007) The challenge posed to children's health by mixtures of toxic waste. Pediatr Clin N Am 54: 155-174. doi: 10.1016/j.pcl.2006.11.009
|
| [6] | Eckel WP, Rabinowitz MB, Foster GD (2011) Discovering unrecognized lead-smelting sites by historical methods. Am J Public Health 91: 625-627. |
| [7] |
Zota AR, Schaider LA, Ettinger AS, et al. (2011) Metal sources and exposures in the homes of young children living near a mining-impacted Superfund site. J Expo Sci Environ Epidemiol 21: 495-505. doi: 10.1038/jes.2011.21
|
| [8] | Falco C (2013) Response to Village re: December 20, 2012 Comments. Illinois EPA, Springfield |
| [9] | Environ International Corporation (2012). Off-Site Soils Design Study. OU4: Off-Site Soils, DePue Site, De Pue, Illinois: Environ International Corporation, On behalf of The DePue Group. Project No. 21-2046C. March 2012. |
| [10] | California Office of Environmental Health Hazards Assessment (COEHHA). (2009) Review Draft Revised California Human Health Screening Level for Lead. Sacramento, California: COEHHA; Available from: http: //oehha.ca.gov/risk/chhsl051809.html. |
| [11] |
Schiffer RB, McDermott MP, Copley C (2001) A multiple-sclerosis cluster associated with a small, north central Illinois community. Arch Environ Health 56: 389-395. doi: 10.1080/00039890109604473
|
| [12] |
Acosta-Saavedra LC, Ma EM, Rodriguez-Kessler T, et al. (2011) Environmental exposure to lead and mercury in Mexican children. Toxicol Mech Methods 21: 656-666. doi: 10.3109/15376516.2011.620997
|
| [13] |
Bar-Sela S, Reingold S, Richter ED (2001) Amyotrophic lateral sclerosis in a battery factory worker exposed to cadmium. Int J Occup Med Environ Health 7: 109-112. doi: 10.1179/oeh.2001.7.2.109
|
| [14] | Braun JM, Kahn RS, Froehlich T, et al. (2006) Exposures to environmental toxicants and attention deficit hyperactivity disorder in US children. Environ Health Persp 114: 1904-1909. |
| [15] |
Chasapis C, Loutsidou AC, Spiliopoulou CA, et al. (2012) Zinc and human health: an update. Arch Toxicol 86: 521-534. doi: 10.1007/s00204-011-0775-1
|
| [16] |
Chen L, Xu B, Liu L, et al. (2011) Cadmium induction of reactive oxygen species activates the mTOR pathway, leading to neuronal cell death. Free Radical Bio Med 50: 624-632. doi: 10.1016/j.freeradbiomed.2010.12.032
|
| [17] |
Choi BY, Jang BG, Kim JH, et al. (2013) Copper/zinc chelation by clioquinol reduces spinal cord white matter damage and behavioral deficits in a murine MOG-induced multiple sclerosis model. Neurobiol Dis 54: 382-391. doi: 10.1016/j.nbd.2013.01.012
|
| [18] |
Farina M, Avila DS, da Rocha JBT, et al. (2013) Metals, oxidative stress and neurodegeneration. Neurochem Int 62: 575-594. doi: 10.1016/j.neuint.2012.12.006
|
| [19] |
Gong G, O'Bryant SE (2010) The arsenic exposure hypothesis for Alzheimer disease. Alzheimer Dis Assoc Disord, 24: 311-316. doi: 10.1097/WAD.0b013e3181d71bc7
|
| [20] |
Gu H, Wei X, Monnot AD, et al. (2011) Lead exposure increases levels of beta-amyloid in the brain and CSF and inhibits LRP1 expression in APP transgenic mice. Neurosci Lett 490: 16-20. doi: 10.1016/j.neulet.2010.12.017
|
| [21] |
Gu H, Robinson G, Hong L, et al. (2012) Increased β amyloid deposition in Tg-SWDI transgenic mouse brain following in vivo lead exposure. Toxicol Lett 213: 211-219. doi: 10.1016/j.toxlet.2012.07.002
|
| [22] |
Guilarte T (2010) APLP1, Alzheimer's-like pathology and neurodegeneration in the frontal cortex of manganese exposed non-human primates. Neurotoxicology 31: 572-574. doi: 10.1016/j.neuro.2010.02.004
|
| [23] |
Kim EA, Kang SK (2010) Occupational neurological disorders in Korea. J Korean Med Soc 25: S26-S35. doi: 10.3346/jkms.2010.25.S.S26
|
| [24] |
Lanphear BP, Hornung R, Khoury J, et al. (2005) Low-level environmental lead exposure and children's intellectual function. Environ Health Persp 113: 894-899. doi: 10.1289/ehp.7688
|
| [25] |
Lassmann H, van Horssen J, Mahad D (2012) Progressive multiple sclerosis. Nat Rev Neurol 8: 647-656. doi: 10.1038/nrneurol.2012.168
|
| [26] |
LeVine SM, Bilgin M, Lynch SG (2013) Iron accumulation in multiple sclerosis. Expert Rev Neurother 13: 247-250. doi: 10.1586/ern.13.14
|
| [27] |
Mates JM, Segura JA, Alonso FJ, et al. (2010) Roles of dioxins and heavy metals in cancer and neurological diseases using ROS-mediated mechanisms. Free Radical Bio Med 49: 1328-1341. doi: 10.1016/j.freeradbiomed.2010.07.028
|
| [28] |
Mazumdar M, Xia W, Hofmann O, et al. (2012) Prenatal lead levels, plasma amyloid beta levels, and gene expression in young adulthood. Environ Health Persp 120: 702-707. doi: 10.1289/ehp.1104474
|
| [29] | Olopade JO, Ponnuru P, Connor JR (2010). Erythropoietin, desferroxamine and iron are protective against vanadium-induced demyelination and oxidative stress in the rat brain Neurotoxicology 31: 582-588. |
| [30] |
Sharma B, Sharma PM (2013) Arsenic toxicity induced endothelial dysfunction and dementia. Toxicol Appl Pharm 273: 180-188. doi: 10.1016/j.taap.2013.07.017
|
| [31] |
Squitti R, Polimanti R, Siotto M, et al. (2013) ATP7B variants as modulators of copper dyshomeostasis in Alzheimer's Disease. NeuroMol Med 15: 515-522. doi: 10.1007/s12017-013-8237-y
|
| [32] |
Suzuki Y, Inoue T, Ra C (2011) Autoimmunity-inducing metals (hg, au, and ag) modulate mast cell signaling, function, and survival. Curr Pharm Design 17: 3805-3814. doi: 10.2174/138161211798357917
|
| [33] |
Taba P (2013) Metals and movement disorders. Curr Opin Neurol 26: 435-441. doi: 10.1097/WCO.0b013e3283629beb
|
| [34] | Tsai CP, Lee CTC (2013) Multiple sclerosis incidence associated with the soil lead and arsenic concentrations in Taiwan. PLoS One 8: 1-6. |
| [35] |
Urrutia P, Aguirre P, Esparza A, et al. (2013) Inflammation alters the expression of DMT1, FPN1 and hepcidin, and it causes iron accumulation in central nervous system cells. J Neurochem 126: 541-549. doi: 10.1111/jnc.12244
|
| [36] | Valera P, Zavattari P, Albanese S, et al. (2013) A correlation study between multiple sclerosis and type 1 diabetes incidences and geochemical data in Europe. Environ Geochem Hlth 10: 79-98. |
| [37] |
Verina T, Schneider JS, Guilarte TR (2013) Manganese exposure induces alpha-synuclein aggregation in the frontal cortex of non-human primates. Toxicol Lett 217: 177-183. doi: 10.1016/j.toxlet.2012.12.006
|
| [38] |
Vitvitsky VM, Sanjay K, Garg K, et al. (2012) Na+ and K+ ion imbalances in Alzheimer's disease. Biochim Biophys Acta 1822: 1671-1681. doi: 10.1016/j.bbadis.2012.07.004
|
| [39] | Wang JD, Huang CC, Hwang YH, et al. (1989) Manganese induced Parkinsonism. Brit J Ind Med 46: 856-859. |
| [40] |
Yorifuji T, Debes F, Weihe P, et al. (2011) Prenatal exposure to lead and cognitive deficit in 7- and 14-year old children in the presence of concomitant exposure to similar molar concentration of methylmercury. Neurotoxicol Teratol 33: 205-211. doi: 10.1016/j.ntt.2010.09.004
|
| [41] |
Pan-Montojo P, Reichmann H (2014) Considerations on the role of environmental toxins in idiopathic Parkinson's disease pathophysiology. Transl Neurodegener 3: 10. doi: 10.1186/2047-9158-3-10
|
| [42] | El-Fawal HAN (2014) Neuroantibody biomarkers. Autoimmune Dis 2014: 1-12. |
| [43] |
Lahiri DK, Maloney B Zawia NH (2009) The LEARn model. Mol Psychiatr 14: 992-1003. doi: 10.1038/mp.2009.82
|
| [44] | World Health Organization (WHO) (2006) Neurological Disorders. World Health Organization, Geneva. |
| [45] | Choo XY, Alukaidey L, White AR, et al. (2013) Neuroinflammation and copper in Alzheimer's disease. Int J Alzheimers Dis 13: 1-12. |
| [46] |
Rosen C, Hansson O, Blennow K, et al. (2013) Fluid biomarkers in Alzheimer's disease: current concepts. Mol Neurodegener 8: 20. doi: 10.1186/1750-1326-8-20
|
| [47] |
Thompson LM (2008) Neurodegeneration: a question of balance. Nature 452: 707-708. doi: 10.1038/452707a
|
| [48] |
Rubinsztein DC (2006) The roles of intracellular protein-degradation pathways in neurodegeneration. Nature 443: 780-6. doi: 10.1038/nature05291
|
| [49] |
Bredesen DE, Rao RV, Mehlen P (2006) Cell death in the nervous system. Nature 443: 796-802. doi: 10.1038/nature05293
|
| [50] |
Calderon-Garcideuenas L, Serrano-Sierra A, Torres-Jardon R, et al. (2013) The impact of environmental metals in young urbanites' brains. Exp Toxicol Pathol 65: 503-511. doi: 10.1016/j.etp.2012.02.006
|
| [51] |
Csavina J, Field J, Taylor MP, et al. (2012) A review on the importance of metals and metalloids in atmospheric dust and aerosol from mining operations. Sci Total Environ 433: 58-73. doi: 10.1016/j.scitotenv.2012.06.013
|
| [52] | Mutter J, Curth A, Naumann J, et al. (2010) Does inorganic mercury play a role in Alzheimer's disease? A systematic review and an integrated molecular mechanism. J Alzheimers Dis 22: 357-374. |
| [53] | Beyersmann D (2002) Effects of carcinogenic metals on gene expression. Toxicol Lett 127(1): 63-68. |
| [54] |
Calabrese EJ, Baldwin LA (2003) Toxicology rethinks its central belief. Nature 421: 691-692. doi: 10.1038/421691a
|
| [55] | Cook R, Calabrese EJ (2006) The importance of hormesis to public health. Environ Health Persp 114: 1631-1635. |
| [56] |
Fukushima S, Kinoshita A, Puatanachokchai R, et al. (2005) Hormesis and dose–response-mediated mechanisms in carcinogenesis. Carcinogenesis 26: 1835-1845. doi: 10.1093/carcin/bgi160
|
| [57] |
Jiang G, Duan W, Xu L, et al. (2009) Biphasic effect of cadmium on cell proliferation in human embryo lung fibroblast cells and its molecular mechanism. Toxicol in Vitro 23: 973-978. doi: 10.1016/j.tiv.2009.06.029
|
| [58] |
Lefcort H, Freedman Z, House S, et al. (2008) Hormetic effects of heavy metals in aquatic snails. EcoHealth 5: 10-17. doi: 10.1007/s10393-008-0158-0
|
| [59] | Schmidt CM (2004) Hormesis effect of trace metals on cultured normal and immortal human mammary cells. Toxicol Ind Health 201: 57-68. |
| [60] | Sandstead HH (1986) A brief history of the influence of trace elements on brain function. Am J Clin Nutr 43: 293-298. |
| [61] |
Fraga CG (2005) Relevance, essentiality and toxicity of trace elements in human health. Mol Aspects Med 26: 235-244. doi: 10.1016/j.mam.2005.07.013
|
| [62] |
Leblanc JC, Guérin T, Noël L, et al. (2005) Dietary exposure estimates of 18 elements from the 1st French total diet study. Food Addit Contam 22: 624-641. doi: 10.1080/02652030500135367
|
| [63] | Duruibe JO, Ogwuegbu MOC, Egwurugwu JN (2007) Heavy metal pollution and human biotoxic effects. Int J Phys Sci 2: 112-118. |
| [64] |
Rainbow PS (2002) Trace metal concentrations in aquatic invertebrates. Environ Pollut 120: 497-507. doi: 10.1016/S0269-7491(02)00238-5
|
| [65] |
Silvera SAN, Rohan TE (2007) Trace elements and cancer risk. Cancer Cause Control 18: 7-27. doi: 10.1007/s10552-006-0057-z
|
| [66] |
Jomova K, Valko M (2011) Advances in metal-induced oxidative stress and human disease. Toxicology 283: 65-87. doi: 10.1016/j.tox.2011.03.001
|
| [67] | Agency for Toxic Substances and Disease Registry: Toxicological Profile for Cobalt. US Department of Health and Human Services, 2004. Available from: http://www.atsdr.cdc.gov.proxy.library.nd.edu/toxprofiles/tp33.pdf. |
| [68] |
Trumbo P, Yates AA, Schliker S, et al. (2001) Dietary reference intakes. J Am Diet Assoc 101: 294-301. doi: 10.1016/S0002-8223(01)00078-5
|
| [69] | Moulton PV, Yang W (2012) Air Pollution, Oxidative Stress, and Alzheimer's Disease. J Environ Public Health 2012: 1-9. |
| [70] |
Rockers PC, Feigla AB, Røttingen J, et al. (2012) Study-design selection criteria in systematic reviews of effectiveness of health systems interventions and reforms: A meta-review. Health Policy 104: 206-214. doi: 10.1016/j.healthpol.2011.12.007
|
| [71] |
Moher D, Liberati AL, Tetzlaff J, et al. (2009) The PRISMA statement for reporting systematic reviews and meta-analyses of studies that evaluate health care interventions: explanation and elaboration. J Clin Epidemiol 62: 1006-1012. doi: 10.1016/j.jclinepi.2009.06.005
|
| [72] |
Khan KS, Kunz R, Kleijnen J, et al. (2003) Five steps to conducting a systematic review. J R Soc Med 96: 118-121. doi: 10.1258/jrsm.96.3.118
|
| [73] | Environmental Protection Agency (EPA): Guidelines for carcinogen risk assessment. Washington, DC Risk Assessment Forum, 2005. Available from: http://www.regulations.gov/#!documentDetail;D=EPA-HQ-OPPT-2010-0877-0021, 2-11. |
| [74] | Environmental Protection Agency (EPA): Endocrine disruptor screening program, weight-of-evidence. Washington, DC, 2011. Available from: http://www.regulations.gov/#!documentDetail;D=EPA-HQ-OPPT-2010-0877-0021, 28. |
| [75] | Hill AB (1965) The environment and disease. P Roy Soc Med 58: 295-300. |
| [76] |
Swaen G, van Amelsvoort L (2009) A weight-of-evidence approach to causal inference. J Clin Epidemiol 62: 270-277. doi: 10.1016/j.jclinepi.2008.06.013
|
| [77] | International Agency for Research on Cancer (IARC): IARC monographs on the evaluation of carcinogenic risks to humans. World Health Organization, 2014. Available from: http://monographs.iarc.fr/ENG/Classification/. |
| [78] |
Phillips CV, Goodman KJ (2004) The missed lessons of Sir Austin Bradford Hill. Epidemiol Perspect Innov 1: 3. doi: 10.1186/1742-5573-1-3
|
| [79] | Institute of Medicine (IOM) Committee on Understanding Premature Birth and Assuring Healthy Outcomes; Behrman RE, Butler AS, editors (2007) Preterm birth: causes, consequences, and prevention. Washington DC: National Academies Press. |
| [80] |
Machamer P, Darden L, Craver CF (2000) Thinking about mechanisms. Philos Sci 67: 1-25. doi: 10.1086/392759
|
| [81] |
Glennan S (2002) Rethinking mechanistic explanation. Philos Sci 69: S342-S353. doi: 10.1086/341857
|
| [82] |
Craver CF (2006) When mechanistic models explain. Synthese 153: 355-376. doi: 10.1007/s11229-006-9097-x
|
| [83] |
Campaner R (2011) Mechanistic causality and counterfactual-manipulative causality. J Epidemiol Commun H 65: 1070-1074. doi: 10.1136/jech.2011.134205
|
| [84] | Brigandt I (2013) Systems biology and the integration of mechanistic explanation and mathematical explanation. Stud Hist Philos Sci C 44: 477-492. |
| [85] |
Garson J (2013) The functional sense of mechanism. Philos Sci 80: 317-333. doi: 10.1086/671173
|
| [86] |
Howick J (2011) Exposing the vanities—and a qualified defense—of mechanistic reasoning in health care decision making. Philos Sci 78: 926-940. doi: 10.1086/662561
|
| [87] |
Leuridan B (2014) The structure of scientific theories, explanation, and unification. A causal–structural account. Br J Philos Sci 65: 717-771. doi: 10.1093/bjps/axt015
|
| [88] |
Thagard P (2003) Pathways to biomedical discovery. Philos Sci 70: 235-254. doi: 10.1086/375465
|
| [89] |
Baetu TM (2012) Genomic Programs as Mechanism Schemas: A Non-Reductionist Interpretation. Brit J Philos Sci 63: 649-671. doi: 10.1093/bjps/axr042
|
| [90] |
Darden L (2008) Thinking again about biological mechanisms. Philos Sci 75: 958-969. doi: 10.1086/594538
|
| [91] | Salmon WC (1989) Four Decades of Scientific Explanation. Minneapolis: University of Minnesota Press. |
| [92] | Feigl H (1970) The ‘Orthodox' View of Theories. In M. Radner and S. Winokur (eds), Minnesota Studies in the Philosophy of Science, Volume IV. Minneapolis: University of Minnesota Press. |
| [93] |
Freiman JA, Chalmers TC, Smith H, et al. (1978) The importance of beta, the type II error and sample size in the design and interpretation of the randomized control trial. N Engl J Med 299: 690-694. doi: 10.1056/NEJM197809282991304
|
| [94] |
Plutynski A (2005) Explanatory unification and the early synthesis. Brit J Philos Sci 56: 595-609. doi: 10.1093/bjps/axi124
|
| [95] |
Cat J (1998) The physicists' debates on unification in physics at the end of the 20th century. Hist Stud Phys Biol 28: 253-299. doi: 10.2307/27757796
|
| [96] |
Nathan MJ (2012) The Varieties of Molecular Explanation. Philos Sci 79: 233-254. doi: 10.1086/664791
|
| [97] | Van Fraassen B (1980) The Scientific Image. New York: Oxford University Press. |
| [98] | Cartwright N (1983) How the Laws of Physics Lie. New York: Oxford University Press. |
| [99] | Woodward J (2003) Making Things Happen: A Theory of Causal Explanation. New York: Oxford University Press, 11. |
| [100] | Waters CK (2007) Causes that Make a Difference. J Philos 104: 551-579. |
| [101] |
Eberhardt F, Scheines R (2007) Interventions and causal inference. Philos Sci 74: 981-995. doi: 10.1086/525638
|
| [102] |
Bokulich A (2012) Distinguishing explanatory from nonexplanatory fictions. Philos Sci 79: 725-737. doi: 10.1086/667991
|
| [103] |
Goodwin W (2013) Quantum Chemistry and Organic Theory. Philos Sci 80: 1159-1169. doi: 10.1086/673734
|
| [104] | Loeb N (2012) Letter of December 17, 2012 to EPA DePue Superfund Site Project Director. Bluhm Legal Clinic, Northwestern Law School. |
| [105] | US EPA (2010) Framework for Incorporating Human Epidemiologic and Incident Data in Health Risk Assessment. Risk Assessment Forum, Office of Pesticides Programs, U.S. Environmental Protection Agency. |
| [106] |
Ou-Yan MH, Van Nostrand WE (2013) The absence of myelin basic protein promotes neuroinflammation and reduces amyloid B-protein accumulation in Tg-5xFAD mice. J Neuroinflammation 10: 134. doi: 10.1186/1742-2094-10-134
|
| [107] |
Butterfield DA, Reed T, Newman SF, et al. (2007) Roles of amyloid β-peptide-associated oxidative stress and brain protein modifications in the pathogenesis of Alzheimer's disease and mild cognitive impairment. Free Radical Bio Med 43: 658-677. doi: 10.1016/j.freeradbiomed.2007.05.037
|
| [108] |
Forster MJ, Dubey A, Dawson KM, et al. (1996) Age-related losses of cognitive function and motor skills in mice are associated with oxidative protein damage in the brain. P Natl Acad Sci USA 93: 4765-4769. doi: 10.1073/pnas.93.10.4765
|
| [109] |
Carney JM, Starke-Reed PE, Oliver CN, et al. (1991) Reversal of age-related increase in brain protein oxidation, decrease in enzyme activity, and loss in temporal and spatial memory by chronic administration of the spin-trapping compound N-tert-butyl-alpha-phenylnitrone. P Natl Acad Sci USA 88: 3633-3636. doi: 10.1073/pnas.88.9.3633
|
| [110] |
Ayton S, Lei P, Bush AI (2013) Metallostasis in Alzheimer's disease. Free Radical Bio Med 62: 76-89. doi: 10.1016/j.freeradbiomed.2012.10.558
|
| [111] |
Faller P, Hureau C, Berthoumieu O (2013) Role of metal ions in the self-assembly of the Alzheimer's amyloid-β peptide. Inorg Chem 52: 12193-12206. doi: 10.1021/ic4003059
|
| [112] |
Greenough MA, Camakaris J, Bush AI (2013) Metal dyshomeostasis and oxidative stress in Alzheimer's disease. Neurochem Int 62: 540-555. doi: 10.1016/j.neuint.2012.08.014
|
| [113] | Jellinger KA (2013) The relevance of metals in the pathophysiology of neurodegeneration, pathological considerations. In International Review of Neurobiology, ed Bhatia KP and Schneider SA, Elsevier, Oxford, 1-48. |
| [114] | Myhre O, Utkilen H, Duale N, et al. (2013) Metal dyshomeostasis and inflammation in Alzheimer's and Parkinson's diseases. Oxid Med Cell Longev 2013: 1-19. |
| [115] |
Obulesu M, Jhansilakshmi M (2014) Neuroinflammation in Alzheimer's disease. Int J Neurosci 124: 227-235. doi: 10.3109/00207454.2013.831852
|
| [116] | Urrutia P, Mena NP, Nunez MT (2014) The interplay between iron accumulation, mitochondrial dysfunction, and inflammation during the execution step of neurodegenerative disorders. Front Pharmacol 38: 1-12. |
| [117] |
Xu PX, Wang SW, Yu XL, et al. (2014) Rutin improves spatial memory in Alzheimer's disease transgenic mice by reducing AB oligomer level and attenuating oxidative stress and neuroinflammation. Behav Brain Res 264: 173-180. doi: 10.1016/j.bbr.2014.02.002
|
| [118] | Zhang W, Yan ZF, Gao JH, et al. (2013) Role and mechanism of microglial activation in iron-induced selective and progressive dopaminergic neurodegeneration. Mol Neurobiol 10: 1-13. |
| [119] |
Nolan YM, Sullivan AM, Toulouse A (2013) Parkinson's disease in the nuclear age of neuroinflammation. Trends Mol Med 19: 187-196. doi: 10.1016/j.molmed.2012.12.003
|
| [120] |
Attar AM, Kharkhaneh A, Etemadifar M, et al. (2012) Serum mercury level and multiple sclerosis. Biol Trace Elem Res 146: 150-153. doi: 10.1007/s12011-011-9239-y
|
| [121] |
Hozumi I, Hasegawa T, Honda A, et al. (2011) Patterns of levels of biological metals in CSF differ among neurodegenerative diseases. J Neurol Sci 303: 95-99. doi: 10.1016/j.jns.2011.01.003
|
| [122] |
Gu H, Wei X, Monnot AD, et al. (2011) Lead exposure increases levels of beta-amyloid in the brain and CSF and inhibits LRP1 expression in APP transgenic mice. Neurosci Lett 490: 16-20. doi: 10.1016/j.neulet.2010.12.017
|
| [123] |
Losy J (2013) Is MS an inflammatory or primary degenerative disease? J Neural Transm 120: 1459-1462. doi: 10.1007/s00702-013-1079-9
|
| [124] |
Brown GC and Neher JJ (2010) Inflammatory neurodegeneration and mechanisms of microglial killing of neurons. Mol Neurobiol 41: 242-247. doi: 10.1007/s12035-010-8105-9
|
| [125] | Serhan CN, Ward PA, Gilroy DW (2010) Acute and chronic inflammation. Fund Inflamm 2010: 1-16. |
| [126] |
Walsh S, Finn DP, Dowd E (2011) Time-course of nigrostriatal neurodegeneration and neuroinflammation in the 6-hydroxydopamine-induced axonal and terminal lesion models of Parkinson's disease in the rat. Neuroscience 175: 251-261. doi: 10.1016/j.neuroscience.2010.12.005
|
| [127] |
Becaria A, Lahiri DK, Bondy SC, et al. (2006) Aluminum and copper in drinking water enhance inflammatory or oxidative events specifically in the brain. J Neuroimmunol 176: 16-23. doi: 10.1016/j.jneuroim.2006.03.025
|
| [128] |
Reinardy HC, Syrett JR, Jeffree RA, et al. (2013) Cobalt-induced genotoxicity in male zebrafish. Aquat Toxicol 126: 224-230. doi: 10.1016/j.aquatox.2012.11.007
|
| [129] |
Roos P M, Vesterberg O, Syversen T, et al. (2013) Metal concentrations in cerebrospinal fluid and blood plasma from patients with amyotrophic lateral sclerosis. Biol Trace Elem Res 151: 159-170. doi: 10.1007/s12011-012-9547-x
|
| [130] |
Paling D, Solanky BS, Riemer F, et al. (2013) Sodium accumulation is associated with disability and a progressive course in multiple sclerosis. Brain 136: 2305-2317 doi: 10.1093/brain/awt149
|
| [131] | Klien K and Godnić-Cvar J (2012) Genotoxicity of metal nanoparticles. Arh Hig Rada Toksikol 63: 133-145. |
| [132] |
Ferreira PC, Kde P, Takayanagui AM, et al. (2008) Aluminum as a risk factor for Alzheimer's disease. Rev Lat Am Enfermagem 16: 151-157. doi: 10.1590/S0104-11692008000100023
|
| [133] | Gillette-Guyonnet S, Andrieu S, Nourhashemi F, et al. (2005) Cognitive impairment and composition of drinking water in women. Am J Clin Nutr 81: 897-902. |
| [134] |
McLachlan DR, Bergeron C, Smith JE, et al. (1996) Risk for neuropathologically confirmed Alzheimer's disease and residual aluminum in municipal drinking water. Neurology 46: 401-405. doi: 10.1212/WNL.46.2.401
|
| [135] |
Forster DP, Newens AJ, Kay DW, et al. (1995) Risk factors in clinically diagnosed presenile dementia of the Alzheimer type. J Epidemiol Community Health 49: 253-258 doi: 10.1136/jech.49.3.253
|
| [136] |
Burns CJ, McIntosh LJ, Mink PJ, et al. (2013) Pesticide Exposure and Neurodevelopmental Outcomes: Review of the Epidemiologic and Animal Studies. J Toxicol Env Heal B 16: 127-283. doi: 10.1080/10937404.2013.783383
|
| [137] | US Environmental Protection Agency (US-EPA): National Priorities List. US-EPA, 2012. Available from: http: //www.epa.gov/superfund/sites/query/queryhtm/nplfin2.htm. |
| [138] |
Gwiazda R, Lucchini R, Smith D (2007) Adequacy and consistency of animal studies to evaluate the neurotoxicity of chronic low-level manganese exposure in humans. J Toxicol Environ Health 70: 594-605. doi: 10.1080/10937400600882897
|
| [139] | Hackam DG, Redelmeier DA (2006) Translation of research evidence from animals to humans. JAMA 296: 1727-1732. |
| [140] |
Apostoli P, Kiss P, Porru S, et al. (1998) Male reproductive toxicity of lead in animals and humans. Occup Environ Med 55: 364-374. doi: 10.1136/oem.55.6.364
|
| [141] |
Alarie, Y (1981) Dose-response analysis in animal studies. Environ Health Persp 42: 9-13. doi: 10.1289/ehp.81429
|
| [142] |
Olson H, Betton G, Robinson D, et al. (2000) Concordance of the toxicity of pharmaceuticals in humans and in animals. Regul Toxicol Pharm 32: 56-67. doi: 10.1006/rtph.2000.1399
|
| [143] | Reagan-Shaw S, Nihal M, Ahmad N (2008) Dose translation from animal to human studies revisited. FASEB J 22: 659-661. |
| [144] |
Shrader-Frechette KS (2008) Evidentiary standards and animal data. Environ Justice 1: 139-144. doi: 10.1089/env.2008.0528
|
| [145] |
Willis AW, Evanoff BA, Lian M, et al. (2010) Metal emissions and urban incident Parkinson Disease. Am J Epidemiol 172: 1357-1363. doi: 10.1093/aje/kwq303
|
| [146] |
Racette BA, McGee-Minnich L, Moerlei SM, et al. (2001) Welding-related parkinsonism. Neurology 56: 8-13. doi: 10.1212/WNL.56.1.8
|
| [147] | Bar-Sela S, Levy M, Westin JB, et al. (1992) Medical findings in nickel-cadmium battery workers. Israel J Med Sci 28: 578-583. |
| [148] |
Miura N (2009) Individual susceptibility to cadmium toxicity and metallothionein gene polymorphisms: with references to current status of occupational cadmium exposure. Ind Health 47: 487-494. doi: 10.2486/indhealth.47.487
|
| [149] | Bellinger DC (2012) A strategy for comparing the contributions of environmental chemicals and other risk factors to neurodevelopment of children. Environ Health Persp 120: 501-507. |
| [150] |
Sanders AP, Miller SK, Nguyen V, et al. (2014) Toxic metal levels in children residing in a smelting-craft village in Vietnam. BMC Pub Health 14: 114. doi: 10.1186/1471-2458-14-114
|
| [151] |
Stein J, Schettler T, Wallinga D, et al. (2002) In harm's way. J Dev Behav Pediatr 23: S13-S22. doi: 10.1097/00004703-200202001-00004
|
| [152] | Mann JM (1997) Medicine and public health, ethics and human rights. The Hastings Center Report 27: 6-13. |
| [153] | Rosenbaum PR (2010) Design of Observational Studies. Springer, New York |
| [154] | Kitcher P (1989) Explanatory unification and the causal structure of the world. Sci Explan 13: 410-505. |
| [155] | Sears ME (2013) Chelation: Harnessing and Enhancing Heavy Metal Detoxification—A Review. ScientificWorldJournal 2013: 1-13. |
| [156] |
Ward RJ, Dexter DT, Crichton RR (2012) Chelating agents for neurodegenerative diseases. Curr Med Chem 19: 2760-2772. doi: 10.2174/092986712800609689
|
| [157] |
Bolognin S, Drago D, Messori L, et al. (2009) Chelation therapy for neurodegenerative diseases. Med Res Rev 29: 547-570. doi: 10.1002/med.20148
|
| [158] | Dang TNT, Lim NKH, Grubman A, et al. (2014) Increased metal content in the TDP-43A315T transgenic mouse model of frontotemporal lobar degeneration and amyotrophic lateral sclerosis. Front Aging Neurosci 6: 15-34. |
| [159] |
Grubman A, White AR, Liddell JR (2014) Mitochondrial metals as a potential therapeutic target in neurodegeneration. Brit J Pharmacol 171: 2159-2173. doi: 10.1111/bph.12513
|
| [160] |
Stephenson E, Nathoo N, Mahjoub Y, et al. (2014) Iron in multiple sclerosis: roles in neurodegeneration and repair. Nat Rev Neurol 10: 459-468. doi: 10.1038/nrneurol.2014.118
|
| [161] | Kanninen KM, Grubman A, Meyerowitz J, et al. (2013) Increased Zinc and Manganese in Parallel with Neurodegeneration, Synaptic Protein Changes and Activation of Akt/GSK3 Signaling in Ovine CLN6 Neuronal Ceroid Lipofuscinosis. PLOS One 8. |
| [162] |
Mitchell SD (2000) Dimensions of Scientific Law. Philos Sci 67: 242-265. doi: 10.1086/392774
|
| [163] | Weber M (2006) The central dogma as a thesis of causal specificity. Hist Phil Life Sci 28: 595-609. |
| [164] |
Hitchcock C (2007) Prevention, preemption, and the principle of sufficient reason. Phil Review 116: 495-532. doi: 10.1215/00318108-2007-012
|
| [165] |
Feldman AL, Johansson AL, Nise G, et al. (2011) Occupational exposure in Parkinsonian disorders: a 43-year prospective cohort study in men. Parkinsonism Relat D 17: 677-682. doi: 10.1016/j.parkreldis.2011.06.009
|
| [166] | Kiesswetter E, Schäper M, Buchta M, et al. (2009) Longitudinal study on potential neurotoxic effects of aluminium: II. Assessment of exposure and neurobehavioral performance of Al welders in the automobile industry over 4 years. Int Arch Occ Env Hea 82: 1191-1210. |
| [167] |
Fored CM, Fryzek JP, Brandt L, et al. (2006) Parkinson's disease and other basal ganglia or movement disorders in a large nationwide cohort of Swedish welders. Occup Environ Med 63: 135-140. doi: 10.1136/oem.2005.022921
|
| [168] |
Santamaria AB, Cushing CA, Antonini JM, et al. (2007) State-of-the-science review: does manganese exposure during welding pose a neurological risk? J Toxicol Env Heal B 10: 417-465. doi: 10.1080/15287390600975004
|
| [169] |
Racette BA, Criswell SR, Lundin JI, et al. (2012) Increased risk of parkinsonism associated with welding exposure. Neurotoxicology 33: 1356-1361. doi: 10.1016/j.neuro.2012.08.011
|
| [170] |
Bowler RM, Roels HA, Nakagawa S, et al. (2007) Dose-effect relationships between manganese exposure and neurological, neuropsychological and pulmonary function in confined space bridge welders. Occup Environ Med 64: 167-177. doi: 10.1136/oem.2006.028761
|
| [171] |
Racette BA, Tabbal SD, Jennings D, et al. (2005) Prevalence of parkinsonism and relationship to exposure in a large sample of Alabama welders. Neurology 64: 230-235. doi: 10.1212/01.WNL.0000149511.19487.44
|
| [172] | National Research Council (2006) Health Risks from Exposure to Low Levels of Ionizing Radiation: BEIR VII Phase 2. Washington, DC: National Academies Press. |
| [173] |
Shrader-Frechette KS (2008) Ideological toxicology: invalid logic, science, ethics about low-dose pollution. Hum Exp Toxicol 27: 647-657. doi: 10.1177/0960327108098491
|
| [174] |
Shrader-Frechette KS (2010) Conceptual analysis and special-interest science. Synthese 177: 449-469. doi: 10.1007/s11229-010-9792-5
|
| [175] | Thayer KA, Melnick R, Burns K, et al. (2005) Fundamental Flaws of Hormesis for Public Health Decisions. Environ Health Persp 113: 1272-1275. |
| [176] | Urani C., Melchioretto P, Fabbri M., et al. (2014). Cadmium impairs p53 activity in hepG2 cells. ISRN Toxicol. Article ID 0976428http: //dx.doi.org/10.1155/2014/976428 |
| [177] | Bal W, Protas AM, Kasprzak KS (2011) Genotoxicity of metal ions. Met Ions Life Sci 8: 319-373. |
| [178] | Shrader-Frechette K (2011). What Will Work: Fighting Climate Change with Renewable Energy, Not Nuclear Power. New York: Oxford University Press. |
Rebecca L. Thomas, Kristin S. Shrader-Frechette. Resolving scientific controversy over smelter risks and neurodegenerative effects of metals[J]. AIMS Environmental Science, 2015, 2(1): 56-86. doi: 10.3934/environsci.2015.1.56
DownLoad: