Modularity of certain products of the Rogers-Ramanujan continued fraction

We study the modularity of the functions of the form $r(\tau)^ar(2\tau)^b$, where $a$ and $b$ are integers with $(a,b)\neq (0,0)$ and $r(\tau)$ is the Rogers-Ramanujan continued fraction, which may be considered as companions to the Ramanujan's function $k(\tau)=r(\tau)r(2\tau)^2$. In particular, we show that under some condition on $a$ and $b$, there are finitely many such functions generating the field of all modular functions on the congruence subgroup $\Gamma_1(10)$. Furthermore, we establish certain arithmetic properties of the function $l(\tau)=r(2\tau)/r(\tau)^2$, which can be used to evaluate these products. We employ the methods of Lee and Park, and some properties of $\eta$-quotients and generalized $\eta$-quotients to prove our results.Comment: 18 pages, comments welcome. arXiv admin note: text overlap with arXiv:2404.0575

Analogue of Ramanujan's function k(τ)k(\tau) for the continued fraction of order six

The Ramanujan's $k(\tau)$ function is defined as $k(\tau)=r(\tau)r(2\tau)^2$, where $r(\tau)$ is the Rogers-Ramanujan continued fraction. Inspired by the recent work of Park (2023) about the analogue of function $k(\tau)$ for the Ramanujan cubic continued fraction, we study certain modular and arithmetic properties of the function $w(\tau) = X(\tau)X(3\tau)$, where $X(\tau)$ is the continued fraction of order six introduced by Vasuki, Bhaskar and Sharath (2010). We consider $w(\tau)$ to be an analogue of $k(\tau)$ for the continued fraction $X(\tau)$

Nutrient Cycling in Forage Production Systems

In most forage production systems, the nutrients needed for plant growth are provided by microbially mediated breakdown and release of plant-available mineral nutrients from dead plant tissues, livestock excreta, soil organic matter, and geochemically bound mineral forms. Even in fertilized forage systems, determining appropriate fertilizer application rates requires a systems approach on the part of the manager (e.g., Di and Cameron, 2000; Rotz et al., 2002). Fertilizer additions are simply one input in the system of inputs, outputs, pools, and fluxes that characterize nutrient cycling in a particular ecosystem

Nitrogen and tillage management for corn following alfalfa

Rotating alfalfa with corn can increase corn yield potential through improved soil physical properties that enhance water infiltration and root extension, a reduction in disease and pest pressure (i.e., corn rootworm), and an enhanced soil microbial community

Nitrogen Cycling in Pasture Grazed by Lactating Dairy Cows

Increasing use of intensive rotational grazing for livestock production in the USA raises questions about the potential for nitrate-N (NO3- N) leaching losses. In grazing experiments with lactating dairy cows at two sites in the Upper Midwest, we monitored milk production, soil NO3-N concentration, and NO3-N leaching. Dietary supplementation increased milk yield, but there was no measurable impact on NO3-N leaching losses. Leaching volumes and NO3-N losses were small on these silt loam soils, even directly under urine patches. The results suggest that NO3-N leaching is not likely to be a problem on fine-textured soils in the Upper Midwest under pastures dominated by deeply-rooted perennial species, as long as N inputs are moderate and animal management does not degrade the pasture

Denitrification under Pastures on Permeable Soils Helps Protect Ground Water Quality

Pastures have been implicated in ground water contamination by nitrate, especially in humid regions with thin or sandy soils (Stout et al., 2000). Significant losses can occur even under low N input, because available N from excreta patches often exceeds plant uptake capacity. Lack of evidence that appreciable nitrate leaching was occurring in established Midwestern USA pastures led us to test the hypothesis that denitrification was preventing or remediating nitrate loading. Higher denitrification rates have been found in the relatively limited number of trials since Ball & Ryden (1984) first reported the significance of this process in pastures

Soil moisture and temperature relationships under fallow in eastern Oregon

Time of stand establishment is a critical factor affecting yields of winter wheat and barley in the fallow-crop rotation areas of the Pacific Northwest. Farmers in this winter-rainfall region are dependent on residual moisture in the seed zone for germination, because significant precipitation does not usually occur until after the optimum planting dates. Moisture is maintained near the soil surface through the summer of the fallow period by the use of a soil or stubble mulch. The rate of moisture loss is relatively low during the summer, but seems to accelerate in late August and September. This loss dries the seed zone and forces either deeper planting to reach adequate moisture or delayed seeding until precipitation re-wets the seed zone. Both practices can result in late, less vigorous stands which have lower yield potential and provide less protection from erosion. The objectives of this study were to: 1) quantify changes in seed zone water content prior to seeding; 2) determine the cause of the accelerated loss; 3) substantiate the effect of planting date on the yield of one variety of winter wheat and one variety of winter barley; 4) investigate the effect of soil temperature on the rate of first and 70% emergence of these species in the field; and 5) develop a means of predicting the average last date of planting after which stand establishment is excessively delayed at one location in eastern Oregon. Although both years were abnormally wet in late summer and fall, significant losses of seed zone water content occurred in 1976. At 6 cm, the loss period occurred in early September; at 9, 12, 15, and 18 cm, the losses occurred in late September. The measured losses were not as great as expected. No significant losses were observed in 1977 because of frequent precipitation. My hypothesis was that increasing nighttime vapor pressure gradients from the moist seed zone to the soil surface develop because of the combination of warm days and cool, clear nights characteristic of late August and September in this area. Larger vapor pressure gradients would cause increased water losses from the profile. However, no correlation was found with calculated vapor pressure gradients, the occurrence of low surface temperatures at night, or average temperature gradients in the upper soil profile. Computer simulation of isothermal liquid flow was used to discern the relative contributions of evaporative losses and of long-term redistribution of water In response to gravitational and potential gradients in the profile. Redistribution accounted for 60% of the water loss in the soil beneath the seed zone from mid-July to early August, and accounted for none of the loss from early August to early September. Planting date had a significant effect on yield of both wheat and barley; the optimum planting dates were late September to early October. On each planting date, the seeds were placed in moist soil and covered with approximately 5 cm of soil with a deep furrow drill, so temperature was the primary factor affecting rate of first and 70% emergence. Regression equations of rate of emergence on average 10-cm soil temperature from planting to emergence were highly significant. The degree days needed for first and 70% emergence for wheat were 149 and 210 using a base temperature of 0.7 and 0.4 C, respectively, and for barley were 92 and 159 using base temperatures of 6.1 and 3.5 C, respectively. Soil temperatures from 1963 to 1977 were used to develop a means of predicting average daily 10-cm soil temperature. Using this long-term average and the regression equations of rate of emergence and stand establishment versus temperature, the average last date to plant and still obtain 70% stand in 14 days was 25 September. If seeding is delayed until 15 October, lower soil temperatures will cause the average days to 70% stand of wheat to approach 22-24 days, while barley will require 24 - 2 9 days to reach 70% stand at these temperatures

Quantitative Glycan-Protein Cross-Linking Mass Spectrometry Using Enrichable Linkers Reveals Extensive Glycan-Mediated Protein Interaction Networks

Protein-protein interactions in the cell membrane are typically mediated by glycans, with terminal sialic acid often involved in these interactions. To probe the nature of the interactions, we developed quantitative cross-linking methods involving the glycans of the glycoproteins and the polypeptide moieties of proteins. We designed and synthesized biotinylated enrichable cross-linkers that were click-tagged to metabolically incorporate azido-sialic acid on cell surface glycans to allow cross-linking of the azido-glycans with lysine residues on proximal polypeptides. The glycopeptide-peptide cross-links (GPx) were enriched using biotin groups through affinity purification with streptavidin resin beads. Workflows using two linkers, one photocleavable and the other disulfide, were developed and applied to reveal the sialic acid-mediated cell-surface protein networks of PNT2 (prostate) cells. Glycopeptide-peptide pairs were identified, with 12000 GPx linked by sialylated glycoforms revealing interactions between source glycoproteins and nearly 700 target proteins. Protein-protein interactions were characterized by as many as 40 peptide pairs, and the extent of the interactions between proteins was prioritized by the number of GPx. Quantitation was performed by counting the number of GPx that identify the protein pairs. Abundant membrane proteins such as ITGB1 yielded an interactome consisting of around 400 other proteins, which were ranked from the most extensive interaction, having the largest number of GPx, to at least one. The interactome was further confirmed separately by published databases. This tool will enhance our understanding of glycosylation on protein-protein interactions and provide new potential targets for therapeutics

Seasonal Distribution of Forage Yield from a Natural Pasture Under Rotational Grazing

A 2-yr grazing study was conducted to quantify forage available daily for cattle intake from a natural pasture managed under rotational grazing. Grazing was initiated around 1 May, and was managed with a rotation length of about 17 d each for cycles 1 and 2, and 30 d each for the rest. In 1994, under adequate moisture conditions, forage availability during 5/1-6/1, 6/1-8/15, 8/15-9/15, and 9/15-10/ 15 was 77, 66, 38 and 14 kg DM ha-1 d-1, respectively, resulting in a total yield of 8580 kg ha-1 in 175 d. In 1995, a prolonged period of dry summer reduced the grazing season to 150 d. Forage supply dropped to 46 kg ha-1 d-1 during 6/1-8/15. Natural pastures in northcentral U.S. have the potential to provide significant amount of forage for 5-6 months per year under rotational grazing, but additional feed may be needed for a month or two during periods of stress