Bounding mean orders of sub-kk-trees of kk-trees

For a $k$-tree $T$, we prove that the maximum local mean order is attained in a $k$-clique of degree $1$ and that it is not more than twice the global mean order. We also bound the global mean order if $T$ has no $k$-cliques of degree $2$ and prove that for large order, the $k$-star attains the minimum global mean order. These results solve the remaining problems of Stephens and Oellermann [J. Graph Theory 88 (2018), 61-79] concerning the mean order of sub-$k$-trees of $k$-trees.Comment: 20 Pages, 6 Figure

A Physiological Examination of the Age-related Decline in Photosynthesis in Picea rubens

Numerous conifer species undergo predictable age-related changes in productivity, photosynthesis and foliar morphology and anatomy. While these phenomena have been demonstrated for many species, the physiological mechanisms controlling them are not well understood. In order to better understand this issue, we examined four possible controls of the age-related decline in photosynthesis in red spruce: stomatal limitation, a decline in investment into photosynthetic capacity, nutrient limitations and a demand-side decline in sink: source relations. We investigated these age-related trends in physiology for juvenile, mid-age and old (mean age ~13, ~54 and ~128 years old) red spruce trees in a multi-cohort stand in Maine. In order to examine stomatal limitations, we examined the diurnal trends in gas exchange parameters, whole tree conductivity and stable carbon isotopes. Photosynthetic capacity parameters (Vcmax and Jmax) and the amount of chlorophyll were examined to evaluate investment in photosynthetic capacity. Amounts of biologically important foliar nutrients (N, P and K) were analyzed to rule out any age-related nutrient deficiencies. Finally, the amount of total non-structural carbohydrates (NSC) were examined in order to better understand the supply and demand of photosynthetic end-product for each age class. All gas exchange parameters were measured in situ, on fully-expanded, current year foliage from the top 1/3 of the canopy using a LI-6400 gas exchange system. The transition from juvenile (sub-canopy) to mid-age (emergent) trees is controlled by increased stomatal limitation and decreased photosynthetic capacity. This was demonstrated by decreases in diurnal trends of gas exchange parameters, photosynthetic capacity parameters (Vcmax and Jmax) and chlorophyll between juvenile and older (mid-age and old) trees. Juvenile trees appear to be operating under a “go for broke strategy” to adapt to intraspecific and interspecific competition. The nutrient limitation hypothesis does not hold true, as there are no age-related difference in foliar nutrient quantities. The transition from mid-age to old trees is marked by a change from source (supply) control to sink (demand) control. Analysis of NSC demonstrates a build-up of photosynthate in old trees, indicating that older, established trees may be using a more conservative life-strategy. As a result, we conclude that the decrease in productivity in old P. rubens is not due to decreased photosynthesis, but to a decreased demand for new carbon compounds. Chapter 2 attempts to distinguish between maturational influences and those due to environmental factors, factors external to the meristem by minimizing the confounding factors of tree size and complexity. Growth and gas exchange measurements were performed on reciprocal grafts in the summer of 2004 using field-grown mid-age and old rootstock and potted juvenile rootstock. Results indicate that factors external to the meristem appear to be the principle influence for age-related changes in photosynthesis and physiology in red spruce

Factorization of Spanning Trees on Feynman Graphs

In order to use the Gaussian representation for propagators in Feynman amplitudes, a representation which is useful to relate string theory and field theory, one has to prove first that each $\alpha$- parameter (where $\alpha$ is the parameter associated to each propagator in the $\alpha$-representation of the Feynman amplitudes) can be replaced by a constant instead of being integrated over and second, prove that this constant can be taken equal for all propagators of a given graph. The first proposition has been proven in one recent letter when the number of propagators is infinite. Here we prove the second one. In order to achieve this, we demonstrate that the sum over the weighted spanning trees of a Feynman graph $G$ can be factorized for disjoint parts of $G$. The same can also be done for cuts on $G$, resulting in a rigorous derivation of the Gaussian representation for super-renormalizable scalar field theories. As a by-product spanning trees on Feynman graphs can be used to define a discretized functional space.Comment: 47 pages, Plain Tex, 3 PostScript figure

A Quantitative Study of Pure Parallel Processes

In this paper, we study the interleaving -- or pure merge -- operator that most often characterizes parallelism in concurrency theory. This operator is a principal cause of the so-called combinatorial explosion that makes very hard - at least from the point of view of computational complexity - the analysis of process behaviours e.g. by model-checking. The originality of our approach is to study this combinatorial explosion phenomenon on average, relying on advanced analytic combinatorics techniques. We study various measures that contribute to a better understanding of the process behaviours represented as plane rooted trees: the number of runs (corresponding to the width of the trees), the expected total size of the trees as well as their overall shape. Two practical outcomes of our quantitative study are also presented: (1) a linear-time algorithm to compute the probability of a concurrent run prefix, and (2) an efficient algorithm for uniform random sampling of concurrent runs. These provide interesting responses to the combinatorial explosion problem

Manejo da biomassa e sustentabilidade nutricional em povoamentos de Eucalyptus spp. em pequenas propriedades rurais

This study was conducted to achieve practical subsidies for nutritional management in Eucalyptus spp. stands in small farms located in Rio Grande do Sul state, Brazil. Different stand ages were evaluated (2, 4, 6 and 8 years old), twenty-four (24) trees were sampled and biomass and nutrients were determined for the following components: leaf, live branch, dead branch, wood and bark from the trunk and root. Biomass and nutrient content in the understory and litter were also estimated. Biomass and nutrients stock in eucalypts were estimated through the product of the mean values for each component, diametric class and number of trees per hectare, and in understory and litter, it was obtained through extrapolation based on the sample unit areas. Utilization efciency from eucalyptus nutrients was estimated through the coefcient of biological utilization (CUB). Considering total nutrients stock in the eucalyptus, nutrients partition followed this order: Ca (40.0%), N (25.6%), K (22.6%), Mg (5.9%), S (3.1%), P (2.8%); and for component: trunk bark (27.5%), leaf (25.1%), wood trunk (22.5%), live branch (14.4%), root (9.4%) and dead branch (1.1%). Litter and understory play an important role in nutrients maintenance in forest ecosystems and accumulate (in average), based on the total stock for eucalyptus, for N, P, K, Ca, Mg and S, respectively, 47.1%, 34.0%, 33.6%, 32.8%, 35.3% and 36.6%. Forest harvesting in young stands leads to loss in wood yield and a higher nutrients relative export (lower CUB). The total eucalyptus biomass harvest above the soil, when compared to the harvest of only the trunk wood, increased the exports of N, P, K, Ca, Mg and S, respectively, in 83%, 152%, 193%, respectively. 445%, 305% and 49%, while the removal of biomass increased only 20%.O estudo foi realizado com o objetivo de subsidiar práticas de manejo nutricional de povoamentos de Eucalyptus spp. em pequenas propriedades rurais no estado do Rio Grande do Sul. Foram avaliados povoamentos de 2, 4, 6 e 8 anos de idade, com amostragem de 24 árvores e determinação da biomassa e nutrientes nos componentes: folha, galho vivo, galho morto, madeira e casca do tronco e raiz. Também foi estimada a biomassa e o conteúdo de nutrientes no sub-bosque e na serapilheira. A biomassa e o estoque de nutrientes no eucalipto foram estimados por meio do produto dos valores médios por componente, classe diamétrica e número de árvores por hectare, e no sub-bosque e serapilheira, por extrapolação com base na área das unidades amostrais. A eficiência de utilização dos nutrientes no eucalipto foi estimada através do coeficiente de utilização biológico (CUB). Considerando-se o estoque total de nutrientes no eucalipto, em média, a partição, por nutriente, obedeceu a seguinte ordem: Ca (40,0%), N (25,6%), K (22,6%), Mg (5,9%), S (3,1%), P (2,8%); e por componente: casca do tronco (27,5%), folha (25,1%), madeira do tronco (22,5%), galho vivo (14,4%), raiz (9,4%) e galho morto (1,1%). A serapilheira e o sub-bosque desempenham importante função na manutenção dos nutrientes no ecossistema florestal, e acumularam em média, em relação ao estoque total no eucalipto, para N, P, K, Ca, Mg e S, respectivamente, 47,1%, 34,0%, 33,6%, 32,8%, 35,3% e 36,6%. A colheita florestal realizada em povoamentos jovens acarreta perda no rendimento de madeira do tronco e maior exportação relativa de nutrientes (menor CUB). A colheita total da biomassa de eucalipto acima do solo, quando comparada com a colheita apenas da madeira do tronco, aumentou a exportação de N, P, K, Ca, Mg e S, respectivamente, em 83%, 152%, 193%, 445%, 305% e 49%, ao passo que a remoção de biomassa aumentou apenas 20%

Introduction to the Sandpile Model

This article is based on a talk given by one of us (EVI) at the conference ``StatPhys-Taipei-1997''. It overviews the exact results in the theory of the sandpile model and discusses shortly yet unsolved problem of calculation of avalanche distribution exponents. The key ingredients include the analogy with the critical reaction-diffusion system, the spanning tree representation of height configurations and the decomposition of the avalanche process into waves of topplings