China's forests cover 208.3 million ha and span a wide range of climates and a large variety of forest types, including tropical, temperate, and boreal forests. However, the variation patterns of fine root (< 2...China's forests cover 208.3 million ha and span a wide range of climates and a large variety of forest types, including tropical, temperate, and boreal forests. However, the variation patterns of fine root (< 2 mm in diameter) biomass, production, and turnover from the south to the north are unclear. This study summarizes fine root biomass (FRB), production (FRP) and turnover rate (FRT) in China's forests as reported by 140 case studies published from 1983 to 2014. The results showed that the mean values of FRB, FRP and FRT in China's forests were 278 g m(-2), 366 g m(-2) a(-1), and 1.19 a(-1), respectively. Compared with other studies at the regional or global scales, FRB in China's forests was lower, FRP was similar to estimates at the global scale, but FRT was much higher. FRB, FRP, and FRT in China's forests increased with increasing mean annual precipitation (MAP), indicating that fine root variables were likely related to MAP, rather than mean annual temperature or latitude. This is possibly due to the small variation in temperature but greater variation in precipitation during the growing season. These findings suggest that spatiotemporal variation in precipitation has a more profound impact on fine root dynamics in China's forests, and this will impact carbon and nutrient cycles driven by root turnover in the future.展开更多
Fine roots play key roles in belowground C cycling in terrestrial ecosystems.Based on their distinct functions,fi ne roots are either absorptive fi ne roots(AFRs)or transport fi ne roots(TFRs).However,the function-bas...Fine roots play key roles in belowground C cycling in terrestrial ecosystems.Based on their distinct functions,fi ne roots are either absorptive fi ne roots(AFRs)or transport fi ne roots(TFRs).However,the function-based fi ne root dynamics of trees and their responses to forest stand properties remain unclear.Here,we studied the dynamics of AFRs and TFRs and their responses to stand conditions and root density in a subtropical montane mixed forest based on a 2-a root window experiment.Mean(±SE)annual production,mortality,and turnover rate of AFRs were 7.87±0.17 m m^(−2)a^(−1),8.13±0.20 m m^(−2)a^(−1)and 2.96±0.24 a^(−1),respectively,compared with 7.09±0.17 m m^(−2)a^(−1),4.59±0.17 m m^(−2)a^(−1),and 2.01±0.22 a^(−1),respectively,for TFRs.The production and mortality of fi ne roots were signifi cantly higher in high root-density sites than in low-root density sites,whereas the turnover of fi ne roots was faster in the low root-density sites.Furthermore,root density had a larger positive eff ect than other environmental factors on TFR production but had no obvious impact on AFR production.Tree species diversity had an apparent positive eff ect on AFR production and was the crucial driver of AFR production,probably due to a complementary eff ect,but had no evident impact on TFR.Both tree density and tree species diversity were positively correlated with the mortality of AFRs and negatively related to the turnover of TFRs,suggesting that higher root density caused stronger competition for rooting space and that plants tend to reduce maintenance costs by decreasing TFR turnover.These fi ndings illustrated the importance of root functional groups in understanding root dynamics and their responses to changes in environmental conditions.展开更多
Fine root turnover is a major pathway for carbon and nutrient cycling in forest ecosystems.However,to estimate fine root turnover,it is important to first understand the fine root dynamic processes associated with soi...Fine root turnover is a major pathway for carbon and nutrient cycling in forest ecosystems.However,to estimate fine root turnover,it is important to first understand the fine root dynamic processes associated with soil resource availability and climate factors.The objectives of this study were:(1)to examine patterns of fine root production and mortality in different seasons and soil depths in the Larix gmelinii and Fraxinus mandshurica plantations,(2)to analyze the correlation of fine root production and mortality with environmental factors such as air temperature,precipitation,soil temperature and available nitrogen,and(3)to estimate fine root turnover.We installed 36 Minirhizotron tubes in six monospecific plots of each species in September 2003 in the Mao’ershan Experimental Forest Station.Minirhizotron sampling was conducted every two weeks from April 2004 to April 2005.We calculated the average fine root length,annual fine root length production and mortality using image data of Minirhizotrons,and estimated fine root turnover using three approaches.Results show that the average growth rate and mortality rate in L.melinii were markedly smaller than in F.mandshurica,and were highest in the surface soil and lowest at the bottom among all the four soil layers.The annual fine root production and mortality in F.mandshurica were significantly higher than in L.gmelinii.The fine root production in spring and summer accounted for 41.7% and 39.7% of the total annual production in F.mandshurica and 24.0% and 51.2% in L.gmelinii.The majority of fine root mortality occurred in spring and summer for F.mandshurica and in summer and autumn for L.gmelinii.The turnover rate was 3.1 a^(-1) for L.gmelinii and 2.7 a^(-1) for F.mandshurica.Multiple regression analysis indicates that climate and soil resource factors together could explain 80% of the variations of the fine root seasonal growth and 95%of the seasonal mortality.In conclusion,fine root production and mortality in L.gmelinii and F.mandshurica have different patterns in different seasons and at different soil depths.Air temperature,precipitation,soil temperature and soil available nitrogen integratively control the dynamics of fine root production,mortality and turnover in both species.展开更多
Soil CO2 efflux, root mass, and root production were investigated in a humid temperate grassland of Japan over a growing season (Apr. to Sep.) of 2005 to reveal seasonal changes of soil CO2 efflux, to separate the r...Soil CO2 efflux, root mass, and root production were investigated in a humid temperate grassland of Japan over a growing season (Apr. to Sep.) of 2005 to reveal seasonal changes of soil CO2 efflux, to separate the respective contributions of root and microbial respiration to the total soil CO2 efttux, and to determine the environmental factors that control soil respiration. Minimal microbial respiration rate was estimated based on the linear regression equations between soil CO2 effiux and root mass at different experimental sites. Soil CO2 efflux, ranging from 4.99 to 16.29 μmol CO2 m^-2 s^-1, depended on the seasonal changes in soil temperature. The root mass at 0-10 cm soil depth was 0.82 and 1.27 kg m^-2 in Apr. and Sep., respectively. The root mass at 0-10 cm soil depth comprised 60% of the total root mass at 0-50 cm soil depth. The root productivity at 0-30 cm depth varied from 8 to 180 g m^-2 month^-1. Microbial and root respiration rates ranged from 1.35 to 5.51 and 2.72 to 12.06μmol CO2 m^-2 s^-1, respectively. The contribution of root respiration to the total soil CO2 efflux averaged 53%, ranging from 33% to 72%. The microbial respiration rate was exponentially related to soil temperature at 10 cm depth (R^2 = 0.9400, P = 0.002, n = 6), and the root respiration rate was linearly related to the root production at 0-30 cm depth (R2 = 0.6561, P = 0.042, n = 6).展开更多
Benefits of repeated air-root-pruning of seedlings when stepping up to progressively larger containers include excellent lateral root distribution immediately below the root collar and an exceptionally fibrous root ba...Benefits of repeated air-root-pruning of seedlings when stepping up to progressively larger containers include excellent lateral root distribution immediately below the root collar and an exceptionally fibrous root ball. To evaluate long-term field performance of repeatedly air-root-pruned container stock, three plantings of swamp white oak (Quercus bicolor Willd.) 10 to 13 years old were located that also included bareroot planting stock. Initial and final stem diameter and height and above-ground green weights were determined on randomly selected trees at each site. On a site with a sandy, excessively drained, high pH soil, trees (age 10) from container stock were 1.5 times taller, 2.3 times larger in dbh, and 2.8 times greater in green weight than trees from bareroot stock which averaged only 2.9 m tall, 3.9 cm dbh, and 16.3 kg green weight. On a site with high clay, poor internal drainage, and frequent flooding, trees (age 12) from container stock were 1.4 times taller, 1.8 times larger in dbh, and 4.1 times greater in green weight than trees from bareroot stock which averaged 4 m tall, 7.3 cm dbh, and 28 kg green weight. On an upland site with deep loess soils, there was a trend for trees (age 13) from container stock to be only slightly larger than trees from bareroot stock with each stock type averaging 9.6 m tall, 20 cm dbh, and 177 kg green weight. Repeated air-root pruning produced lateral roots immediately below the root collar that resulted in large container stock with large well-balanced root systems that were competitive on harsh or less than ideal oak sites. Although the process is relatively labor intensive, propagation of repeatedly air-root-pruned container stock is readily adaptable internationally to locally available sources of organic matter and open-bottom containers.展开更多
The belowground biomass is represented by coarse and fine roots. Concentrated in the superficial horizons of the soil, the fine roots play a crucial role in the functioning of a forest ecosystem. However, studies on t...The belowground biomass is represented by coarse and fine roots. Concentrated in the superficial horizons of the soil, the fine roots play a crucial role in the functioning of a forest ecosystem. However, studies on their dynamics in natural forests are almost non-existent in the Republic of Congo. Here, we estimated the biomass, production, turnover and fine root lifespan of two forest strata of a semi-deciduous forest: the <i><span style="font-family:Verdana;">Gilbertiodendron dewevrei</span></i><span style="font-family:Verdana;"> (De Wild.) J. Léonard forest (GF) and the mixed forest (MF) of land. The ingrowth cores method was used to estimate the biomass, production, turnover and lifespan of fine roots. The results of this study revealed that the biomass, production and fine root turnover of the two forest strata studied significantly decreased with increasing soil depth, with an increase in lifespan. The annual fine root biomass of GF (2284.50 ± 37.62 <img src="Edit_990c94b6-013e-4e21-90df-d1388dc0e65f.png" alt="" /></span><span style="font-family:Verdana;"> and 1034.61 ± 14.52 <img src="Edit_dff42540-5a2f-413b-8620-cb500e9961e2.png" alt="" /></span><span style="font-family:Verdana;">) was slightly lower than that of MF (2430.07 ± 40.68 <img src="Edit_66800589-8460-4c37-83b2-2df0f335d75d.png" alt="" /></span><span style="font-family:Verdana;"> and 1043.10 ± 11.75 <img src="Edit_c22f255e-d910-4b49-a6a4-033516044362.png" alt="" /></span><span style="font-family:Verdana;">) in the 0-15 cm and 15-30 cm horizons, respectively. The annual production of fine roots from these latter horizons was respectively 1300.19 ± 32.17 <img src="Edit_5482204b-8e9e-476a-907d-0865bf3a1c99.png" alt="" /></span><span style="font-family:Verdana;"> and 539.18 ± 11.55 <img src="Edit_65a2856e-5322-4fc9-b42a-3ba1176fa992.png" alt="" /></span><span style="font-family:Verdana;"> in GF and 1362.24 ± 39.59 <img src="Edit_9802e464-658d-48eb-9b57-8e746c3e8ef4.png" alt="" /></span><span style="font-family:Verdana;"> and 492.95 ± 14.38 <img src="Edit_51413fca-930c-45b9-a385-2b55d4d2bac8.png" alt="" /></span><span style="font-family:Verdana;"> in the MF. Root turnover was higher in the GF (1.68 ± 0.05 <img src="Edit_ce9d780c-6a46-46c4-aad2-653309318e29.png" alt="" /></span><span style="font-family:Verdana;"> and 1.35 ± 0.03 <img src="Edit_d66d8b7b-c608-4398-9441-e85547f03dea.png" alt="" /></span><span style="font-family:Verdana;">) than in the MF (1.57 ± 0.05 <img src="Edit_cb79094f-88a0-401c-a3e7-06eedb2cef9a.png" alt="" /></span><span style="font-family:Verdana;"> and 1.13 ± 0.02 <img src="Edit_e4f9b6d7-2e2e-44d5-8662-862b8f8ff80e.png" alt="" /></span><span style="font-family:Verdana;">). The lifespan of fine roots increased with the depth of the soil. The difference in fine root dynamics observed between the forest strata studied was influenced by the Evenness index and the above-ground biomass.</span>展开更多
This article originates from the observation that field lines are drawn using distinctive rules in magnetic field and electrostatic fields. It aims at reconciliating the definitions of these fields and thus reaching a...This article originates from the observation that field lines are drawn using distinctive rules in magnetic field and electrostatic fields. It aims at reconciliating the definitions of these fields and thus reaching a consensus on the interpretation of field lines. Our unified field definition combines three orthogonal vectors and a unique scalar value. Field lines are then defined as isovalue lines of the scalar value, rendering it simpler to interpret in both field types. Specific to our field definition is the use of square root of vector’s cross product so that all vectors have the same physical unit. This enhanced field definition also enables a more efficient calculation of Biot-Savart law. This article is the first of a series allowing the drawing of isovalue contour lines.展开更多
An independent set in a graph G is a set of pairwise non-adjacent vertices. The independence polynomial of G is the polynomial AΣx^(|A|), where the sum is over all independent sets A of G. In 1987, Alavi,Malde, Schwe...An independent set in a graph G is a set of pairwise non-adjacent vertices. The independence polynomial of G is the polynomial AΣx^(|A|), where the sum is over all independent sets A of G. In 1987, Alavi,Malde, Schwenk and Erd os conjectured that the independence polynomial of any tree or forest is unimodal.Although this unimodality conjecture has attracted many researchers’ attention, it is still open. Recently, Basit and Galvin even asked a much stronger question whether the independence polynomial of every tree is ordered log-concave. Note that if a polynomial has only negative real zeros then it is ordered log-concave and unimodal.In this paper, we observe real-rootedness of independence polynomials of rooted products of graphs. We find some trees whose rooted product preserves real-rootedness of independence polynomials. In consequence, starting from any graph whose independence polynomial has only real zeros, we can obtain an infinite family of graphs whose independence polynomials have only real zeros. In particular, applying it to trees or forests, we obtain that their independence polynomials are unimodal and ordered log-concave.展开更多
基金supported by Grants from the National Key Research and Development Program of China(2016YFD06004040604)the Natural Science Foundation of Heilongjiang Province(No.C2016004)
文摘China's forests cover 208.3 million ha and span a wide range of climates and a large variety of forest types, including tropical, temperate, and boreal forests. However, the variation patterns of fine root (< 2 mm in diameter) biomass, production, and turnover from the south to the north are unclear. This study summarizes fine root biomass (FRB), production (FRP) and turnover rate (FRT) in China's forests as reported by 140 case studies published from 1983 to 2014. The results showed that the mean values of FRB, FRP and FRT in China's forests were 278 g m(-2), 366 g m(-2) a(-1), and 1.19 a(-1), respectively. Compared with other studies at the regional or global scales, FRB in China's forests was lower, FRP was similar to estimates at the global scale, but FRT was much higher. FRB, FRP, and FRT in China's forests increased with increasing mean annual precipitation (MAP), indicating that fine root variables were likely related to MAP, rather than mean annual temperature or latitude. This is possibly due to the small variation in temperature but greater variation in precipitation during the growing season. These findings suggest that spatiotemporal variation in precipitation has a more profound impact on fine root dynamics in China's forests, and this will impact carbon and nutrient cycles driven by root turnover in the future.
基金supported by the National Natural Science Foundation of China(Grant No.3217159931870465).
文摘Fine roots play key roles in belowground C cycling in terrestrial ecosystems.Based on their distinct functions,fi ne roots are either absorptive fi ne roots(AFRs)or transport fi ne roots(TFRs).However,the function-based fi ne root dynamics of trees and their responses to forest stand properties remain unclear.Here,we studied the dynamics of AFRs and TFRs and their responses to stand conditions and root density in a subtropical montane mixed forest based on a 2-a root window experiment.Mean(±SE)annual production,mortality,and turnover rate of AFRs were 7.87±0.17 m m^(−2)a^(−1),8.13±0.20 m m^(−2)a^(−1)and 2.96±0.24 a^(−1),respectively,compared with 7.09±0.17 m m^(−2)a^(−1),4.59±0.17 m m^(−2)a^(−1),and 2.01±0.22 a^(−1),respectively,for TFRs.The production and mortality of fi ne roots were signifi cantly higher in high root-density sites than in low-root density sites,whereas the turnover of fi ne roots was faster in the low root-density sites.Furthermore,root density had a larger positive eff ect than other environmental factors on TFR production but had no obvious impact on AFR production.Tree species diversity had an apparent positive eff ect on AFR production and was the crucial driver of AFR production,probably due to a complementary eff ect,but had no evident impact on TFR.Both tree density and tree species diversity were positively correlated with the mortality of AFRs and negatively related to the turnover of TFRs,suggesting that higher root density caused stronger competition for rooting space and that plants tend to reduce maintenance costs by decreasing TFR turnover.These fi ndings illustrated the importance of root functional groups in understanding root dynamics and their responses to changes in environmental conditions.
基金The funding for this research was provided by theNationalNatural Science Foundation of China(Grant No.30130160).
文摘Fine root turnover is a major pathway for carbon and nutrient cycling in forest ecosystems.However,to estimate fine root turnover,it is important to first understand the fine root dynamic processes associated with soil resource availability and climate factors.The objectives of this study were:(1)to examine patterns of fine root production and mortality in different seasons and soil depths in the Larix gmelinii and Fraxinus mandshurica plantations,(2)to analyze the correlation of fine root production and mortality with environmental factors such as air temperature,precipitation,soil temperature and available nitrogen,and(3)to estimate fine root turnover.We installed 36 Minirhizotron tubes in six monospecific plots of each species in September 2003 in the Mao’ershan Experimental Forest Station.Minirhizotron sampling was conducted every two weeks from April 2004 to April 2005.We calculated the average fine root length,annual fine root length production and mortality using image data of Minirhizotrons,and estimated fine root turnover using three approaches.Results show that the average growth rate and mortality rate in L.melinii were markedly smaller than in F.mandshurica,and were highest in the surface soil and lowest at the bottom among all the four soil layers.The annual fine root production and mortality in F.mandshurica were significantly higher than in L.gmelinii.The fine root production in spring and summer accounted for 41.7% and 39.7% of the total annual production in F.mandshurica and 24.0% and 51.2% in L.gmelinii.The majority of fine root mortality occurred in spring and summer for F.mandshurica and in summer and autumn for L.gmelinii.The turnover rate was 3.1 a^(-1) for L.gmelinii and 2.7 a^(-1) for F.mandshurica.Multiple regression analysis indicates that climate and soil resource factors together could explain 80% of the variations of the fine root seasonal growth and 95%of the seasonal mortality.In conclusion,fine root production and mortality in L.gmelinii and F.mandshurica have different patterns in different seasons and at different soil depths.Air temperature,precipitation,soil temperature and soil available nitrogen integratively control the dynamics of fine root production,mortality and turnover in both species.
基金supported by the National Natural Science Foundation of China (Nos.30670342 and 30870408)the JilinProvincial Natural Science Funds for Distinguished Young Scholars of China (No.20060105).
文摘Soil CO2 efflux, root mass, and root production were investigated in a humid temperate grassland of Japan over a growing season (Apr. to Sep.) of 2005 to reveal seasonal changes of soil CO2 efflux, to separate the respective contributions of root and microbial respiration to the total soil CO2 efttux, and to determine the environmental factors that control soil respiration. Minimal microbial respiration rate was estimated based on the linear regression equations between soil CO2 effiux and root mass at different experimental sites. Soil CO2 efflux, ranging from 4.99 to 16.29 μmol CO2 m^-2 s^-1, depended on the seasonal changes in soil temperature. The root mass at 0-10 cm soil depth was 0.82 and 1.27 kg m^-2 in Apr. and Sep., respectively. The root mass at 0-10 cm soil depth comprised 60% of the total root mass at 0-50 cm soil depth. The root productivity at 0-30 cm depth varied from 8 to 180 g m^-2 month^-1. Microbial and root respiration rates ranged from 1.35 to 5.51 and 2.72 to 12.06μmol CO2 m^-2 s^-1, respectively. The contribution of root respiration to the total soil CO2 efflux averaged 53%, ranging from 33% to 72%. The microbial respiration rate was exponentially related to soil temperature at 10 cm depth (R^2 = 0.9400, P = 0.002, n = 6), and the root respiration rate was linearly related to the root production at 0-30 cm depth (R2 = 0.6561, P = 0.042, n = 6).
文摘Benefits of repeated air-root-pruning of seedlings when stepping up to progressively larger containers include excellent lateral root distribution immediately below the root collar and an exceptionally fibrous root ball. To evaluate long-term field performance of repeatedly air-root-pruned container stock, three plantings of swamp white oak (Quercus bicolor Willd.) 10 to 13 years old were located that also included bareroot planting stock. Initial and final stem diameter and height and above-ground green weights were determined on randomly selected trees at each site. On a site with a sandy, excessively drained, high pH soil, trees (age 10) from container stock were 1.5 times taller, 2.3 times larger in dbh, and 2.8 times greater in green weight than trees from bareroot stock which averaged only 2.9 m tall, 3.9 cm dbh, and 16.3 kg green weight. On a site with high clay, poor internal drainage, and frequent flooding, trees (age 12) from container stock were 1.4 times taller, 1.8 times larger in dbh, and 4.1 times greater in green weight than trees from bareroot stock which averaged 4 m tall, 7.3 cm dbh, and 28 kg green weight. On an upland site with deep loess soils, there was a trend for trees (age 13) from container stock to be only slightly larger than trees from bareroot stock with each stock type averaging 9.6 m tall, 20 cm dbh, and 177 kg green weight. Repeated air-root pruning produced lateral roots immediately below the root collar that resulted in large container stock with large well-balanced root systems that were competitive on harsh or less than ideal oak sites. Although the process is relatively labor intensive, propagation of repeatedly air-root-pruned container stock is readily adaptable internationally to locally available sources of organic matter and open-bottom containers.
文摘The belowground biomass is represented by coarse and fine roots. Concentrated in the superficial horizons of the soil, the fine roots play a crucial role in the functioning of a forest ecosystem. However, studies on their dynamics in natural forests are almost non-existent in the Republic of Congo. Here, we estimated the biomass, production, turnover and fine root lifespan of two forest strata of a semi-deciduous forest: the <i><span style="font-family:Verdana;">Gilbertiodendron dewevrei</span></i><span style="font-family:Verdana;"> (De Wild.) J. Léonard forest (GF) and the mixed forest (MF) of land. The ingrowth cores method was used to estimate the biomass, production, turnover and lifespan of fine roots. The results of this study revealed that the biomass, production and fine root turnover of the two forest strata studied significantly decreased with increasing soil depth, with an increase in lifespan. The annual fine root biomass of GF (2284.50 ± 37.62 <img src="Edit_990c94b6-013e-4e21-90df-d1388dc0e65f.png" alt="" /></span><span style="font-family:Verdana;"> and 1034.61 ± 14.52 <img src="Edit_dff42540-5a2f-413b-8620-cb500e9961e2.png" alt="" /></span><span style="font-family:Verdana;">) was slightly lower than that of MF (2430.07 ± 40.68 <img src="Edit_66800589-8460-4c37-83b2-2df0f335d75d.png" alt="" /></span><span style="font-family:Verdana;"> and 1043.10 ± 11.75 <img src="Edit_c22f255e-d910-4b49-a6a4-033516044362.png" alt="" /></span><span style="font-family:Verdana;">) in the 0-15 cm and 15-30 cm horizons, respectively. The annual production of fine roots from these latter horizons was respectively 1300.19 ± 32.17 <img src="Edit_5482204b-8e9e-476a-907d-0865bf3a1c99.png" alt="" /></span><span style="font-family:Verdana;"> and 539.18 ± 11.55 <img src="Edit_65a2856e-5322-4fc9-b42a-3ba1176fa992.png" alt="" /></span><span style="font-family:Verdana;"> in GF and 1362.24 ± 39.59 <img src="Edit_9802e464-658d-48eb-9b57-8e746c3e8ef4.png" alt="" /></span><span style="font-family:Verdana;"> and 492.95 ± 14.38 <img src="Edit_51413fca-930c-45b9-a385-2b55d4d2bac8.png" alt="" /></span><span style="font-family:Verdana;"> in the MF. Root turnover was higher in the GF (1.68 ± 0.05 <img src="Edit_ce9d780c-6a46-46c4-aad2-653309318e29.png" alt="" /></span><span style="font-family:Verdana;"> and 1.35 ± 0.03 <img src="Edit_d66d8b7b-c608-4398-9441-e85547f03dea.png" alt="" /></span><span style="font-family:Verdana;">) than in the MF (1.57 ± 0.05 <img src="Edit_cb79094f-88a0-401c-a3e7-06eedb2cef9a.png" alt="" /></span><span style="font-family:Verdana;"> and 1.13 ± 0.02 <img src="Edit_e4f9b6d7-2e2e-44d5-8662-862b8f8ff80e.png" alt="" /></span><span style="font-family:Verdana;">). The lifespan of fine roots increased with the depth of the soil. The difference in fine root dynamics observed between the forest strata studied was influenced by the Evenness index and the above-ground biomass.</span>
文摘This article originates from the observation that field lines are drawn using distinctive rules in magnetic field and electrostatic fields. It aims at reconciliating the definitions of these fields and thus reaching a consensus on the interpretation of field lines. Our unified field definition combines three orthogonal vectors and a unique scalar value. Field lines are then defined as isovalue lines of the scalar value, rendering it simpler to interpret in both field types. Specific to our field definition is the use of square root of vector’s cross product so that all vectors have the same physical unit. This enhanced field definition also enables a more efficient calculation of Biot-Savart law. This article is the first of a series allowing the drawing of isovalue contour lines.
基金supported by the National Natural Science Foundation of China (Nos.11971206, 12022105)the Natural Science Foundation of Distinguished Young Scholars of Jiangsu Province (No.BK20200048)Postgraduate Research Practice&Innovation Program of Jiangsu Province (No.KYCX21-2565)。
文摘An independent set in a graph G is a set of pairwise non-adjacent vertices. The independence polynomial of G is the polynomial AΣx^(|A|), where the sum is over all independent sets A of G. In 1987, Alavi,Malde, Schwenk and Erd os conjectured that the independence polynomial of any tree or forest is unimodal.Although this unimodality conjecture has attracted many researchers’ attention, it is still open. Recently, Basit and Galvin even asked a much stronger question whether the independence polynomial of every tree is ordered log-concave. Note that if a polynomial has only negative real zeros then it is ordered log-concave and unimodal.In this paper, we observe real-rootedness of independence polynomials of rooted products of graphs. We find some trees whose rooted product preserves real-rootedness of independence polynomials. In consequence, starting from any graph whose independence polynomial has only real zeros, we can obtain an infinite family of graphs whose independence polynomials have only real zeros. In particular, applying it to trees or forests, we obtain that their independence polynomials are unimodal and ordered log-concave.