Super congruences and Euler numbers

Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\binom{2k}{k}/2^k=(-1)^{(p-1)/2}-p^2E_{p-3} (mod p^3),$$ $$\sum_{k=1}^{(p-1)/2}\binom{2k}{k}/k=(-1)^{(p+1)/2}8/3*pE_{p-3} (mod p^2),$$ $$\sum_{k=0}^{(p-1)/2}\binom{2k}{k}^2/16^k=(-1)^{(p-1)/2}+p^2E_{p-3} (mod p^3)$$, where E_0,E_1,E_2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for $\pi^2$, $\pi^{-2}$ and the constant $K:=\sum_{k>0}(k/3)/k^2$ (with (-) the Jacobi symbol), two of which are $$\sum_{k=1}^\infty(10k-3)8^k/(k^3\binom{2k}{k}^2\binom{3k}{k})=\pi^2/2$$ and $$\sum_{k>0}(15k-4)(-27)^{k-1}/(k^3\binom{2k}{k}^2\binom{3k}k)=K.$

Decadal changes in fire frequencies shift tree communities and functional traits

Global change has resulted in chronic shifts in fire regimes. Variability in the sensitivity of tree communities to multi-decadal changes in fire regimes is critical to anticipating shifts in ecosystem structure and function, yet remains poorly understood. Here, we address the overall effects of fire on tree communities and the factors controlling their sensitivity in 29 sites that experienced multi-decadal alterations in fire frequencies in savanna and forest ecosystems across tropical and temperate regions. Fire had a strong overall effect on tree communities, with an average fire frequency (one fire every three years) reducing stem density by 48% and basal area by 53% after 50 years, relative to unburned plots. The largest changes occurred in savanna ecosystems and in sites with strong wet seasons or strong dry seasons, pointing to fire characteristics and species composition as important. Analyses of functional traits highlighted the impact of fire-driven changes in soil nutrients because frequent burning favoured trees with low biomass nitrogen and phosphorus content, and with more efficient nitrogen acquisition through ectomycorrhizal symbioses. Taken together, the response of trees to altered fire frequencies depends both on climatic and vegetation determinants of fire behaviour and tree growth, and the coupling between fire-driven nutrient losses and plant traits

Green and animal manure use in organic field crop systems

Dual-use cover/green manure (CGM) crops and animal manure are used to supply nitrogen (N) and phosphorus (P) to organically grown field crops. A comprehensive review of previous research was conducted to identify how CGM crops and animal manure have been used to meet N and P needs of organic field crops, and to identify knowledge gaps to direct future research efforts. Results indicate that: (a) CGM crops are used to provide N to subsequent cash crops in rotations; (b) CGM-supplied N generally can meet field crop needs in warm, humid regions but is insufficient for organic grain crops grown in cool and sub-humid regions; (c) adoption of conservation tillage practices can create or exacerbate N deficiencies; (d) excess N and P can result where animal manures are accessible if application rates are not carefully managed; and (e) integrating animal grazing into organic field crop systems has potential benefits but is generally not practiced. Work is needed to better understand the mechanisms governing the release of N by CGM crops to subsequent cash crops, and the legacy effects of animal manure applications in cool and sub-humid regions. The benefits and synergies that can occur by combining targeted animal grazing and CGMs on soil N, P, and other nutrients should be investigated. Improved communication and networking among researchers can aid efforts to solve soil fertility challenges faced by organic farmers when growing field crops in North America and elsewhere

Precipitation gradients drive high tree species turnover in the woodlands of eastern and southern Africa

Savannas cover one-fifth of the Earth's surface, harbour substantial biodiversity, and provide a broad range of ecosystem services to hundreds of millions of people. The community composition of trees in tropical moist forests varies with climate, but whether the same processes structure communities in disturbance-driven savannas remains relatively unknown. We investigate how biodiversity is structured over large environmental and disturbance gradients in woodlands of eastern and southern Africa. We use tree inventory data from the Socio-Ecological Observatory for Studying African Woodlands (SEOSAW) network, covering 755 ha in a total of 6780 plots across nine countries of eastern and southern Africa, to investigate how alpha, beta, and phylogenetic diversity varies across environmental and disturbance gradients. We find strong climate-richness patterns, with precipitation playing a primary role in determining patterns of tree richness and high turnover across these savannas. Savannas with greater rainfall contain more tree species, suggesting that low water availability places distributional limits on species, creating the observed climate-richness patterns. Both fire and herbivory have minimal effects on tree diversity, despite their role in determining savanna distribution and structure. High turnover of tree species, genera, and families is similar to turnover in seasonally dry tropical forests of the Americas, suggesting this is a feature of semiarid tree floras. The greater richness and phylogenetic diversity of wetter plots shows that broad-scale ecological patterns apply to disturbance-driven savanna systems. High taxonomic turnover suggests that savannas from across the regional rainfall gradient should be protected if we are to maximise the conservation of unique tree communities

Carbon losses from deforestation and widespread degradation offset by extensive growth in African woodlands

Degradation—the loss of carbon stored in intact woodland—is very difficult to measure over large areas. Here, the authors show that carbon emissions from degradation in African woodlands greatly exceed those from deforestation, but are happening alongside widespread increases in biomass in remote areas

Long-Term Vegetation Change in Central Africa: The Need for an Integrated Management Framework for Forests and Savannas

peer reviewedTropical forests and savannas are the main biomes in sub-Saharan Africa, covering most of the continent. Collectively they offer important habitat for biodiversity and provide multiple ecosystem services. Considering their global importance and the multiple sustainability challenges they face in the era of the Anthropocene, this chapter undertakes a comprehensive analysis of the past, present, and future vegetation patterns in central African forests and savannas. Past changes in climate, vegetation, land use, and human activity have affected the distribution of forests and savannas across central Africa. Currently, forests form a continuous block across the wet and moist areas of central Africa, and are characterized by high tree cover (>90% tree cover). Savannas and woodlands have lower tree cover (<40% tree cover), are found in drier sites in the north and south of the region, and are maintained by frequent fires. Recent tree cover loss (2000–2015) has been more important for forests than for savannas, which, however, reportedly experienced woody encroachment. Future cropland expansion is expected to have a strong impact on savannas, while the extent of climatic impacts depends on the actual scenario. We finally identify some of the policy implications for restoring ecosystems, expanding protected areas, and designing sustainable ecosystem management approaches in the region