scholarly journals Large Crown Root Number Improves Topsoil Foraging and Phosphorus Acquisition

2018 ◽  
Vol 177 (1) ◽  
pp. 90-104 ◽  
Author(s):  
Baoru Sun ◽  
Yingzhi Gao ◽  
Jonathan P. Lynch
Planta ◽  
2009 ◽  
Vol 230 (4) ◽  
pp. 599-610 ◽  
Author(s):  
Jie Xiong ◽  
Han Lu ◽  
Kaixing Lu ◽  
Yuxing Duan ◽  
Lingyao An ◽  
...  

2014 ◽  
Vol 166 (2) ◽  
pp. 581-589 ◽  
Author(s):  
P. Saengwilai ◽  
X. Tian ◽  
J. P. Lynch

2019 ◽  
Vol 241 ◽  
pp. 107562 ◽  
Author(s):  
Zhigang Liu ◽  
Yang Zhao ◽  
Song Guo ◽  
Shuai Cheng ◽  
Yanjie Guan ◽  
...  

2021 ◽  
Author(s):  
Nicolas Honvault ◽  
David Houben ◽  
Stéphane Firmin ◽  
Hacène Meglouli ◽  
Frédéric Laruelle ◽  
...  

Author(s):  
Robert F. Brown

AbstractLet $$\phi :X \multimap Y$$ ϕ : X ⊸ Y be an n-valued map of connected finite polyhedra and let $$a \in Y$$ a ∈ Y . Then, $$x \in X$$ x ∈ X is a root of $$\phi $$ ϕ at a if $$a \in \phi (x)$$ a ∈ ϕ ( x ) . The Nielsen root number $$N(\phi : a)$$ N ( ϕ : a ) is a lower bound for the number of roots at a of any n-valued map homotopic to $$\phi $$ ϕ . We prove that if X and Y are compact, connected triangulated manifolds without boundary, of the same dimension, then given $$\epsilon > 0$$ ϵ > 0 , there is an n-valued map $$\psi $$ ψ homotopic to $$\phi $$ ϕ within Hausdorff distance $$\epsilon $$ ϵ of $$\phi $$ ϕ such that $$\psi $$ ψ has finitely many roots at a. We conjecture that if X and Y are q-manifolds without boundary, $$q \ne 2$$ q ≠ 2 , then there is an n-valued map homotopic to $$\phi $$ ϕ that has $$N(\phi : a)$$ N ( ϕ : a ) roots at a. We verify the conjecture when $$X = Y$$ X = Y is a Lie group by employing a fixed point result of Schirmer. As an application, we calculate the Nielsen root numbers of linear n-valued maps of tori.


2019 ◽  
Vol 66 (2) ◽  
pp. 168-181 ◽  
Author(s):  
Djamel Houassine ◽  
Mourad Latati ◽  
Nazih Y. Rebouh ◽  
Frédéric Gérard

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