The Discrete Horizontal Complex on Lattice Space

We define discrete total differential forms on lattice space by. changing coefficients of discrete differential forms from functions only of n to functions also of dependent variables un and their partial differences. And the discrete exterior derivative extends to be discrete total differential map which is also nilpotent. Then a discrete horizontal complex can be derived and be proved to be exact by constructing homotopy operators.

Noncommutative Differential Calculus and Its Application on Discrete Spaces

We present the noncommutative differential calculus on the function space of the infinite set and construct a homotopy operator to prove the analogue of the Poincare lemma for the difference complex. Then the horizontal and vertical complexes are introduced with the total differential map and vertical exterior derivative. As the application of the differential calculus, we derive the schemes with the conservation of symplecticity and energy for Hamiltonian system and a two-dimensional integral models with infinite sequence of conserved currents. Then an Euler-Lagrange cohomology with symplectic structure-preserving is given in the discrete classical mechanics.

The coordinated impact of forest internal structural complexity and tree species diversity on forest productivity across forest biomes

Forest structural complexity can mediate the light and water distribution within forest canopies,and has a direct impact on forest biodiversity and carbon storage capability.It is believed that increases in forest structural complexity can enhance tree species diversity and forest productivity,but inconsistent relationships among them have been reported.Here,we quantified forest structural complexity in three aspects(i.e.,horizontal,vertical,and internal structural complexity)from unmanned aerial vehicle light detection and ranging data,and investigated their correlations with tree species diversity and forest productivity by incorporating field measurements in three forest biomes with large latitude gradients in China.Our results show that internal structural complexity had a stronger correlation(correlation coefficient=0.85)with tree species richness than horizontal structural complexity(correlation coefficient=-0.16)and vertical structural complexity(correlation coefficient=0.61),and it was the only forest structural complexity attribute having significant correlations with both tree species richness and tree species evenness.A strong scale effect was observed in the correlations among forest structural complexity,tree species diversity,and forest productivity.Moreover,forest internal structural complexity had a tight positive coordinated contribution with tree species diversity to forest productivity through structure equation model analysis,while horizontal and vertical structural complexity attributes have insignificant or weaker coordinated effects than internal structural complexity,which indicated that the neglect of forest internal structural complexity might partially lead to the current inconsistent observations among forest structural complexity,tree species diversity,and forest productivity.The results of this study can provide a new angle to understand the observed inconsistent correlations among forest structural complexity,tree species diversity,and forest productivity.

Laplacians on the holomorphic tangent bundle of a Kaehler manifold

Let M be a connected complex manifold endowed with a Hermitian metric g.In this paper,the complex horizontal and vertical Laplacians associated with the induced Hermitian metric <·,·>on the holomorphic tangent bundle T 1,0M of M are defined,and their explicit expressions are obtained.Using the complex horizontal and vertical Laplacians associated with the Hermitian metric <·,·>on T 1,0M,we obtain a vanishing theorem of holomorphic horizontal p forms which are compactly supported in T 1,0M under the condition that g is a Kaehler metric on M.