Addition formulas exist in trigonometric functions.Double-angle and half-angle formulas can be derived from these formulas.Moreover,the relation equation between the trigonometric function and the hyperbolic function ...Addition formulas exist in trigonometric functions.Double-angle and half-angle formulas can be derived from these formulas.Moreover,the relation equation between the trigonometric function and the hyperbolic function can be derived using an imaginary number.The inverse hyperbolic function arsinher(r)■ro 1/√1+t^(2)dt p1tt2 dt is similar to the inverse trigonometric function arcsiner(r)■ro 1/√1+t^(2)dt p1t2 dt,such as the second degree of a polynomial and the constant term 1,except for the sign−and+.Such an analogy holds not only when the degree of the polynomial is 2,but also for higher degrees.As such,a function exists with respect to the leaf function through the imaginary number i,such that the hyperbolic function exists with respect to the trigonometric function through this imaginary number.In this study,we refer to this function as the hyperbolic leaf function.By making such a definition,the relation equation between the leaf function and the hyperbolic leaf function makes it possible to easily derive various formulas,such as addition formulas of hyperbolic leaf functions based on the addition formulas of leaf functions.Using the addition formulas,we can also derive the double-angle and half-angle formulas.We then verify the consistency of these formulas by constructing graphs and numerical data.展开更多
Leaf functional traits are adaptations that enable plants to live under different environmental conditions. This study aims to evaluate the differences in leaf functional traits between red and green leaves of two eve...Leaf functional traits are adaptations that enable plants to live under different environmental conditions. This study aims to evaluate the differences in leaf functional traits between red and green leaves of two evergreen shrubs Photinia × fraseri and Osrnanthus fragrans. Specific areas of red leaves are higher than that of green leaves in both species. Thus, the material investment per unit area and per lamina of red leaves is significantly lower than that of green leaves, implying an utmost effort of red leaves to increase light capture and use efficiency because of their low leafchlorophyll concentration. The higher petiole length of green leaves compared with that of red leaves indicates that adult green leaves may have large fractional biomass allocation to support the lamina structures in capturing light with maximum efficiency and obtaining a high growth rate. The high range of the phenotypic plasticity of leaf size, leaf thickness, single-leaf wet and dry weights, and leaf moisture of green leaves may be beneficial in achieving efficient control of water loss and nutrient deprivation. The high range of phenotypic plasticity of leaf chlorophyll concentration of red leaves may be advantageous in increasing resource （especially light） capture anduse efficiency because this leaf type is juvenile in the growth stage and has low leaf-chlorophyll concentration.展开更多
Variations in leaf functional traits of Abies georgei var. smithii at 3700, 3900, 4100, 4300, and 4390 m altitude were investigated in 15 typical plots in the Southeastern Tibetan Plateau. In each plot, three seedling...Variations in leaf functional traits of Abies georgei var. smithii at 3700, 3900, 4100, 4300, and 4390 m altitude were investigated in 15 typical plots in the Southeastern Tibetan Plateau. In each plot, three seedlings were selected, of which functional leaves in current-year sunny branches were chosen for the measurement of morphological, photosynthetic, and physiological and biochemical characteristics, and their variations were analyzed. Results showed that significant variations existed among the leaf functional traits of A. georgei var. smithii along the altitudinal gradient, as well as their physiological adaption indicators. Leaf area decreased, while the mass per area and thickness of leaf increased at an altitude above 4,100 m. The maxima of pigment, total nitrogen concentration, net photosynthesis rate during light-saturated, and when water use efficiency appeared at 4100 m altitude. In addition, A. georgei var. smithii seedlings regulated the activities of superoxide dismutase and ascorbate peroxidase to resist abiotic stress under 4100 m altitude. Meanwhile, malondialdehyde concentration and the dark respiration rate rapidly increased, which indicates that A. georgei var. smithii seedlingssuffered from heavy abiotic stress from 4100 m to 4390 m altitude. Basing on variations in leaf functional traits along the altitude gradient, we inferred that 4100 m altitude was the suitable region for A. georgei var. smithii growth in the Sygera Mountain. Moreover, the harsh environment was the main limiting factor for A. georgei var. smithii population expansion to high altitude.展开更多
A lemniscate is a curve defined by two foci,F_(1) and F_(2).If the distance between the focal points of F_(1)−F_(2) is 2a(a:constant),then any point P on the lemniscate curve satisfy the equation PF_(1)·PF_(2)=a^...A lemniscate is a curve defined by two foci,F_(1) and F_(2).If the distance between the focal points of F_(1)−F_(2) is 2a(a:constant),then any point P on the lemniscate curve satisfy the equation PF_(1)·PF_(2)=a^(2).Jacob Bernoulli first described the lemniscate in 1694.The Fagnano discovered the double angle formula of the lemniscate(1718).The Euler extended the Fagnano’s formula to a more general addition theorem(1751).The lemniscate function was subsequently proposed by Gauss around the year 1800.These insights were summarized by Jacobi as the theory of elliptic functions.A leaf function is an extended lemniscate function.Some formulas of leaf functions have been presented in previous papers;these included the addition theorem of this function and its application to nonlinear equations.In this paper,the geometrical properties of leaf functions at n=2 and the geometric relation between the angle θ and lemniscate arc length l are presented using the lemniscate curve.The relationship between the leaf functions sleaf_(2)(l)and cleaf_(2)(l)is derived using the geometrical properties of the lemniscate,similarity of triangles,and the Pythagorean theorem.In the literature,the relation equation for sleaf_(2)(l)and cleaf_(2)(l)(or the lemniscate functions,sl(l)and cl(l))has been derived analytically;however,it is not derived geometrically.展开更多
This study aims to determine the differences in leaf functional traits and phenotypic plasticity of leaf functional traits between exotic and native Compositae plant species. Leaf width of exotic plants was significan...This study aims to determine the differences in leaf functional traits and phenotypic plasticity of leaf functional traits between exotic and native Compositae plant species. Leaf width of exotic plants was significantly lower than that of native species. Leaf length, specific leaf area(SLA), single-leaf wet and dry weights, leaf moisture, and leaf thickness of exotic plants were also lower than those of native species but not significantly. The leaf shape index of exotic plants was higher than that of native species but not significantly. This implies that the relatively low leaf construction cost for exotic plants may play an important role in the success of their invasions. The higher leaf shape index and lower leaf width of exotic plants can enhance the efficiency of resource capture(especially sunlight capture) via adjustments to leaf shape and size, thereby increasing the survival of exotic plants. The plasticity indices of single-leaf wet weight and leaf thickness of exotic plants were significantly lower than those of native species. The lower phenotypic plasticity of single-leaf wet weight and leaf thickness of exotic plants may be the result of a cost to plasticity. That is, if the plasticity is too high, the fitness of plant species might be reduced sharply under unfavorable environments. Thus, lower plasticity of leaf functional traits may compensate for the negative impact of adverse environments and stabilize leaf construction costs for exotic plants. Moreover, reduced phenotypic plasticity might be one of the key competitive strategies by which exotic plants successfully invade new habitats. Overall, exotic plants did not always exhibit higher values of leaf functional traits or increased phenotypic plasticity of leaf functional traits compared with native species.展开更多
Aims To explore the pattern of the leaf functional traits of shrub species along a latitudinal gradient in eastern China and determine the driv-ing factors of leaf trait variation at a large scale.Methods We investiga...Aims To explore the pattern of the leaf functional traits of shrub species along a latitudinal gradient in eastern China and determine the driv-ing factors of leaf trait variation at a large scale.Methods We investigated the leaf thickness(LT),leaf area(LA),specific leaf area(SLA)and leaf dry mass content(LDMC)of 185 shrub species from 13 sites across eastern China.The trends of these four-leaf traits were ana-lyzed with respect to latitude,and the differences between different life forms(e.g.,evergreen and deciduous)and habitats(e.g.,understory and typical)were compared.We quantified the effects of the plant life forms and environmental factors on the leaf traits via mixed-model analyses.Important Findings The LT and LA decreased,whilst and the LDMC increased,as the latitude increased,and significant differences in these traits were observed between the different plant life forms.The LT and LA were smaller,whereas the SLA and LDMC were larger in decidu-ous shrubs than in evergreen shrubs.Among the different habitats,the LA and SLA were larger,while the LDMC was smaller in under-story shrubs than in typical shrub species.These results indicate that typical shrub species are better adapted to drier environments,as indicated by a reduced LT and increased LDMC.Furthermore,general linear models showed that variations in the four-leaf traits with respect to latitude were mainly caused by a shift in plant life forms.展开更多
文摘Addition formulas exist in trigonometric functions.Double-angle and half-angle formulas can be derived from these formulas.Moreover,the relation equation between the trigonometric function and the hyperbolic function can be derived using an imaginary number.The inverse hyperbolic function arsinher(r)■ro 1/√1+t^(2)dt p1tt2 dt is similar to the inverse trigonometric function arcsiner(r)■ro 1/√1+t^(2)dt p1t2 dt,such as the second degree of a polynomial and the constant term 1,except for the sign−and+.Such an analogy holds not only when the degree of the polynomial is 2,but also for higher degrees.As such,a function exists with respect to the leaf function through the imaginary number i,such that the hyperbolic function exists with respect to the trigonometric function through this imaginary number.In this study,we refer to this function as the hyperbolic leaf function.By making such a definition,the relation equation between the leaf function and the hyperbolic leaf function makes it possible to easily derive various formulas,such as addition formulas of hyperbolic leaf functions based on the addition formulas of leaf functions.Using the addition formulas,we can also derive the double-angle and half-angle formulas.We then verify the consistency of these formulas by constructing graphs and numerical data.
基金financially supported by the National Natural Science Foundation of China(31300343)Natural Science Foundation of Jiangsu Province,China(BK20130500)Jiangsu Collaborative Innovation Center of Technology and Material of Water Treatment
文摘Leaf functional traits are adaptations that enable plants to live under different environmental conditions. This study aims to evaluate the differences in leaf functional traits between red and green leaves of two evergreen shrubs Photinia × fraseri and Osrnanthus fragrans. Specific areas of red leaves are higher than that of green leaves in both species. Thus, the material investment per unit area and per lamina of red leaves is significantly lower than that of green leaves, implying an utmost effort of red leaves to increase light capture and use efficiency because of their low leafchlorophyll concentration. The higher petiole length of green leaves compared with that of red leaves indicates that adult green leaves may have large fractional biomass allocation to support the lamina structures in capturing light with maximum efficiency and obtaining a high growth rate. The high range of the phenotypic plasticity of leaf size, leaf thickness, single-leaf wet and dry weights, and leaf moisture of green leaves may be beneficial in achieving efficient control of water loss and nutrient deprivation. The high range of phenotypic plasticity of leaf chlorophyll concentration of red leaves may be advantageous in increasing resource （especially light） capture anduse efficiency because this leaf type is juvenile in the growth stage and has low leaf-chlorophyll concentration.
基金supported by the Tibetan Natural Scientific Foundation of China(2015ZR13-28)the Doctoral Scientific Research Foundation(STSD-2)+2 种基金Tibetan Linzhi National Forest Ecological Research Station(2012-LYPTDW-016)Promotion Plan of Plateau Basic Ecological Academic Team Abilitysupported by CFERN&GENE Award funds on ecological paper
文摘Variations in leaf functional traits of Abies georgei var. smithii at 3700, 3900, 4100, 4300, and 4390 m altitude were investigated in 15 typical plots in the Southeastern Tibetan Plateau. In each plot, three seedlings were selected, of which functional leaves in current-year sunny branches were chosen for the measurement of morphological, photosynthetic, and physiological and biochemical characteristics, and their variations were analyzed. Results showed that significant variations existed among the leaf functional traits of A. georgei var. smithii along the altitudinal gradient, as well as their physiological adaption indicators. Leaf area decreased, while the mass per area and thickness of leaf increased at an altitude above 4,100 m. The maxima of pigment, total nitrogen concentration, net photosynthesis rate during light-saturated, and when water use efficiency appeared at 4100 m altitude. In addition, A. georgei var. smithii seedlings regulated the activities of superoxide dismutase and ascorbate peroxidase to resist abiotic stress under 4100 m altitude. Meanwhile, malondialdehyde concentration and the dark respiration rate rapidly increased, which indicates that A. georgei var. smithii seedlingssuffered from heavy abiotic stress from 4100 m to 4390 m altitude. Basing on variations in leaf functional traits along the altitude gradient, we inferred that 4100 m altitude was the suitable region for A. georgei var. smithii growth in the Sygera Mountain. Moreover, the harsh environment was the main limiting factor for A. georgei var. smithii population expansion to high altitude.
基金supported by Daido University research Grants(2020).
文摘A lemniscate is a curve defined by two foci,F_(1) and F_(2).If the distance between the focal points of F_(1)−F_(2) is 2a(a:constant),then any point P on the lemniscate curve satisfy the equation PF_(1)·PF_(2)=a^(2).Jacob Bernoulli first described the lemniscate in 1694.The Fagnano discovered the double angle formula of the lemniscate(1718).The Euler extended the Fagnano’s formula to a more general addition theorem(1751).The lemniscate function was subsequently proposed by Gauss around the year 1800.These insights were summarized by Jacobi as the theory of elliptic functions.A leaf function is an extended lemniscate function.Some formulas of leaf functions have been presented in previous papers;these included the addition theorem of this function and its application to nonlinear equations.In this paper,the geometrical properties of leaf functions at n=2 and the geometric relation between the angle θ and lemniscate arc length l are presented using the lemniscate curve.The relationship between the leaf functions sleaf_(2)(l)and cleaf_(2)(l)is derived using the geometrical properties of the lemniscate,similarity of triangles,and the Pythagorean theorem.In the literature,the relation equation for sleaf_(2)(l)and cleaf_(2)(l)(or the lemniscate functions,sl(l)and cl(l))has been derived analytically;however,it is not derived geometrically.
基金Project(31300343)supported by the National Natural Science Foundation of ChinaProject(Y20160023)supported by Open Science Research Fund of State Key Laboratory of Soil and Sustainable Agriculture,Institute of Soil Science,Chinese Academy of Sciences,Chinasupported by Jiangsu Collaborative Innovation Center of Technology and Material of Water Treatment,China
文摘This study aims to determine the differences in leaf functional traits and phenotypic plasticity of leaf functional traits between exotic and native Compositae plant species. Leaf width of exotic plants was significantly lower than that of native species. Leaf length, specific leaf area(SLA), single-leaf wet and dry weights, leaf moisture, and leaf thickness of exotic plants were also lower than those of native species but not significantly. The leaf shape index of exotic plants was higher than that of native species but not significantly. This implies that the relatively low leaf construction cost for exotic plants may play an important role in the success of their invasions. The higher leaf shape index and lower leaf width of exotic plants can enhance the efficiency of resource capture(especially sunlight capture) via adjustments to leaf shape and size, thereby increasing the survival of exotic plants. The plasticity indices of single-leaf wet weight and leaf thickness of exotic plants were significantly lower than those of native species. The lower phenotypic plasticity of single-leaf wet weight and leaf thickness of exotic plants may be the result of a cost to plasticity. That is, if the plasticity is too high, the fitness of plant species might be reduced sharply under unfavorable environments. Thus, lower plasticity of leaf functional traits may compensate for the negative impact of adverse environments and stabilize leaf construction costs for exotic plants. Moreover, reduced phenotypic plasticity might be one of the key competitive strategies by which exotic plants successfully invade new habitats. Overall, exotic plants did not always exhibit higher values of leaf functional traits or increased phenotypic plasticity of leaf functional traits compared with native species.
基金Grants were received from the National Natural Science Foundation of China(31470498 and 31330012)the Chinese National Basic Research Program(2015CB954201 and 2014CB954303)the Strategic Priority Research Program of the Chinese Academy of Sciences(XDA05050701).
文摘Aims To explore the pattern of the leaf functional traits of shrub species along a latitudinal gradient in eastern China and determine the driv-ing factors of leaf trait variation at a large scale.Methods We investigated the leaf thickness(LT),leaf area(LA),specific leaf area(SLA)and leaf dry mass content(LDMC)of 185 shrub species from 13 sites across eastern China.The trends of these four-leaf traits were ana-lyzed with respect to latitude,and the differences between different life forms(e.g.,evergreen and deciduous)and habitats(e.g.,understory and typical)were compared.We quantified the effects of the plant life forms and environmental factors on the leaf traits via mixed-model analyses.Important Findings The LT and LA decreased,whilst and the LDMC increased,as the latitude increased,and significant differences in these traits were observed between the different plant life forms.The LT and LA were smaller,whereas the SLA and LDMC were larger in decidu-ous shrubs than in evergreen shrubs.Among the different habitats,the LA and SLA were larger,while the LDMC was smaller in under-story shrubs than in typical shrub species.These results indicate that typical shrub species are better adapted to drier environments,as indicated by a reduced LT and increased LDMC.Furthermore,general linear models showed that variations in the four-leaf traits with respect to latitude were mainly caused by a shift in plant life forms.
