Title: Integral representations of rank two false theta functions and their modularity properties
Abstract False theta functions form a family of functions with intriguing modular properties and connections to mock modular forms. In this paper, we take the first step towards investigating modular transformations of higher rank false theta functions, following the example of higher depth mock modular forms. In particular, we prove that under quite general conditions, a rank two false theta function is determined in terms of iterated, holomorphic, Eichler-type integrals. This provides a new method for examining their modular properties and we apply it in a variety of situations where rank two false theta functions arise. We first consider generic parafermion characters of vertex algebras of type $$A_2$$ A 2 and $$B_2$$ B 2 . This requires a fairly non-trivial analysis of Fourier coefficients of meromorphic Jacobi forms of negative index, which is of independent interest. Then we discuss modularity of rank two false theta functions coming from superconformal Schur indices. Lastly, we analyze $${\hat{Z}}$$ Z ^ -invariants of Gukov, Pei, Putrov, and Vafa for certain plumbing $$\mathtt{H}$$ H -graphs. Along the way, our method clarifies previous results on depth two quantum modularity.
Folsom, Amanda(
, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences)
This article is in commemoration of Ramanujan's election as Fellow of The Royal Society 100 years ago, as celebrated at the October 2018 scientific meeting at the Royal Society in London. Ramanujan's last letter to Hardy, written shortly after his election, surrounds his mock theta functions. While these functions have been of great importance and interest in the decades following Ramanujan's death in 1920, it was unclear how exactly they fit into the theory of modular forms—Dyson called this ‘a challenge for the future’ at another centenary conference in Illinois in 1987, honouring the 100th anniversary of Ramanujan's birth. In the early 2000s, Zwegers finally recognized that Ramanujan had discovered glimpses of special families of non-holomorphic modular forms, which we now know to be Bruinier and Funke's harmonic Maass forms from 2004, the holomorphic parts of which are called mock modular forms. As of a few years ago, a fundamental question from Ramanujan's last letter remained, on a certain asymptotic relationship between mock theta functions and ordinary modular forms. The author, with Ono and Rhoades, revisited Ramanujan's asymptotic claim, and established a connection between mock theta functions and quantum modular forms, which were not defined until 90 years later inmore »2010 by Zagier. Here, we bring together past and present, and study the relationships between mock modular forms and quantum modular forms, with Ramanujan's mock theta functions as motivation. In particular, we highlight recent work of Bringmann–Rolen, Choi–Lim–Rhoades and Griffin–Ono–Rolen in our discussion. This article is largely expository, but not exclusively: we also establish a new interpretation of Ramanujan's radial asymptotic limits in the subject of topology. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.« less
Hamana, Takashi; Shirasaki, Masato; Miyazaki, Satoshi; Hikage, Chiaki; Oguri, Masamune; More, Surhud; Armstrong, Robert; Leauthaud, Alexie; Mandelbaum, Rachel; Miyatake, Hironao; et al(
, Publications of the Astronomical Society of Japan)
Abstract We present measurements of cosmic shear two-point correlation functions (TPCFs) from Hyper Suprime-Cam Subaru Strategic Program (HSC) first-year data, and derive cosmological constraints based on a blind analysis. The HSC first-year shape catalog is divided into four tomographic redshift bins ranging from $z=0.3$ to 1.5 with equal widths of $\Delta z =0.3$. The unweighted galaxy number densities in each tomographic bin are 5.9, 5.9, 4.3, and $2.4\:$arcmin$^{-2}$ from the lowest to highest redshifts, respectively. We adopt the standard TPCF estimators, $\xi _\pm$, for our cosmological analysis, given that we find no evidence of significant B-mode shear. The TPCFs are detected at high significance for all 10 combinations of auto- and cross-tomographic bins over a wide angular range, yielding a total signal-to-noise ratio of 19 in the angular ranges adopted in the cosmological analysis, $7^{\prime }<\theta <56^{\prime }$ for $\xi _+$ and $28^{\prime }<\theta <178^{\prime }$ for $\xi _-$. We perform the standard Bayesian likelihood analysis for cosmological inference from the measured cosmic shear TPCFs, including contributions from intrinsic alignment of galaxies as well as systematic effects from PSF model errors, shear calibration uncertainty, and source redshift distribution errors. We adopt a covariance matrix derived from realistic mock catalogs constructedmore »from full-sky gravitational lensing simulations that fully account for survey geometry and measurement noise. For a flat $\Lambda$ cold dark matter model, we find $S\,_8 \equiv \sigma _8\sqrt{\Omega _{\rm m}/0.3}=0.804_{-0.029}^{+0.032}$, and $\Omega _{\rm m}=0.346_{-0.100}^{+0.052}$. We carefully check the robustness of the cosmological results against astrophysical modeling uncertainties and systematic uncertainties in measurements, and find that none of them has a significant impact on the cosmological constraints.« less
We derive a holomorphic anomaly equation for the Vafa-Wittenpartition function for twisted four-dimensional \mathcal{N} =4 𝒩 = 4 super Yang-Mills theory on \mathbb{CP}^{2} ℂ ℙ 2 for the gauge group SO(3) S O ( 3 ) from the path integral of the effective theory on the Coulomb branch.The holomorphic kernel of this equation, which receives contributionsonly from the instantons, is not modular but ‘mock modular’. Thepartition function has correct modular properties expected from S S -dualityonly after including the anomalous nonholomorphic boundary contributionsfrom anti-instantons. Using M-theory duality, we relate this phenomenonto the holomorphic anomaly of the elliptic genus of a two-dimensionalnoncompact sigma model and compute it independently in two dimensions.The anomaly both in four and in two dimensions can be traced to atopological term in the effective action of six-dimensional (2,0) ( 2 , 0 ) theory on the tensor branch. We consider generalizations to othermanifolds and other gauge groups to show that mock modularity is genericand essential for exhibiting duality when the relevant field space isnoncompact.
Hussain, Mir Zaman; Hamilton, Stephen; Robertson, G. Philip; Basso, Bruno(
)
Abstract
Excessive phosphorus (P) applications to croplands can contribute to eutrophication of surface waters through surface runoff and subsurface (leaching) losses. We analyzed leaching losses of total dissolved P (TDP) from no-till corn, hybrid poplar (Populus nigra X P. maximowiczii), switchgrass (Panicum virgatum), miscanthus (Miscanthus giganteus), native grasses, and restored prairie, all planted in 2008 on former cropland in Michigan, USA. All crops except corn (13 kg P ha−1 year−1) were grown without P fertilization. Biomass was harvested at the end of each growing season except for poplar. Soil water at 1.2 m depth was sampled weekly to biweekly for TDP determination during March–November 2009–2016 using tension lysimeters. Soil test P (0–25 cm depth) was measured every autumn. Soil water TDP concentrations were usually below levels where eutrophication of surface waters is frequently observed (> 0.02 mg L−1) but often higher than in deep groundwater or nearby streams and lakes. Rates of P leaching, estimated from measured concentrations and modeled drainage, did not differ statistically among cropping systems across years; 7-year cropping system means ranged from 0.035 to 0.072 kg P ha−1 year−1 with large interannual variation. Leached P was positively related to STP, which decreased over the 7 years in all systems. These results indicate that both P-fertilized and unfertilized cropping systems may
leach legacy P from past cropland management.
Methods
Experimental details The Biofuel Cropping System Experiment (BCSE) is located at the W.K. Kellogg Biological Station (KBS) (42.3956° N, 85.3749° W; elevation 288 m asl) in southwestern Michigan, USA. This site is a part of the Great Lakes Bioenergy Research Center (www.glbrc.org) and is a Long-term Ecological Research site (www.lter.kbs.msu.edu). Soils are mesic Typic Hapludalfs developed on glacial outwash54 with high sand content (76% in the upper 150 cm) intermixed with silt-rich loess in the upper 50 cm55. The water table lies approximately 12–14 m below the surface. The climate is humid temperate with a mean annual air temperature of 9.1 °C and annual precipitation of 1005 mm, 511 mm of which falls between May and September (1981–2010)56,57. The BCSE was established as a randomized complete block design in 2008 on preexisting farmland. Prior to BCSE establishment, the field was used for grain crop and alfalfa (Medicago sativa L.) production for several decades. Between 2003 and 2007, the field received a total of ~ 300 kg P ha−1 as manure, and the southern half, which contains one of four replicate plots, received an additional 206 kg P ha−1 as inorganic fertilizer. The experimental design consists of five randomized blocks each containing one replicate plot (28 by 40 m) of 10 cropping systems (treatments) (Supplementary Fig. S1; also see Sanford et al.58). Block 5 is not included in the present study. Details on experimental design and site history are provided in Robertson and Hamilton57 and Gelfand et al.59. Leaching of P is analyzed in six of the cropping systems: (i) continuous no-till corn, (ii) switchgrass, (iii) miscanthus, (iv) a mixture of five species of native grasses, (v) a restored native prairie containing 18 plant species (Supplementary Table S1), and (vi) hybrid poplar. Agronomic management Phenological cameras and field observations indicated that the perennial herbaceous crops emerged each year between mid-April and mid-May. Corn was planted each year in early May. Herbaceous crops were harvested at the end of each growing season with the timing depending on weather: between October and November for corn and between November and December for herbaceous perennial crops. Corn stover was harvested shortly after corn grain, leaving approximately 10 cm height of stubble above the ground. The poplar was harvested only once, as the culmination of a 6-year rotation, in the winter of 2013–2014. Leaf emergence and senescence based on daily phenological images indicated the beginning and end of the poplar growing season, respectively, in each year. Application of inorganic fertilizers to the different crops followed a management approach typical for the region (Table 1). Corn was fertilized with 13 kg P ha−1 year−1 as starter fertilizer (N-P-K of 19-17-0) at the time of planting and an additional 33 kg P ha−1 year−1 was added as superphosphate in spring 2015. Corn also received N fertilizer around the time of planting and in mid-June at typical rates for the region (Table 1). No P fertilizer was applied to the perennial grassland or poplar systems (Table 1). All perennial grasses (except restored prairie) were provided 56 kg N ha−1 year−1 of N fertilizer in early summer between 2010 and 2016; an additional 77 kg N ha−1 was applied to miscanthus in 2009. Poplar was fertilized once with 157 kg N ha−1 in 2010 after the canopy had closed. Sampling of subsurface soil water and soil for P determination Subsurface soil water samples were collected beneath the root zone (1.2 m depth) using samplers installed at approximately 20 cm into the unconsolidated sand of 2Bt2 and 2E/Bt horizons (soils at the site are described in Crum and Collins54). Soil water was collected from two kinds of samplers: Prenart samplers constructed of Teflon and silica (http://www.prenart.dk/soil-water-samplers/) in replicate blocks 1 and 2 and Eijkelkamp ceramic samplers (http://www.eijkelkamp.com) in blocks 3 and 4 (Supplementary Fig. S1). The samplers were installed in 2008 at an angle using a hydraulic corer, with the sampling tubes buried underground within the plots and the sampler located about 9 m from the plot edge. There were no consistent differences in TDP concentrations between the two sampler types. Beginning in the 2009 growing season, subsurface soil water was sampled at weekly to biweekly intervals during non-frozen periods (April–November) by applying 50 kPa of vacuum to each sampler for 24 h, during which the extracted water was collected in glass bottles. Samples were filtered using different filter types (all 0.45 µm pore size) depending on the volume of leachate collected: 33-mm dia. cellulose acetate membrane filters when volumes were less than 50 mL; and 47-mm dia. Supor 450 polyethersulfone membrane filters for larger volumes. Total dissolved phosphorus (TDP) in water samples was analyzed by persulfate digestion of filtered samples to convert all phosphorus forms to soluble reactive phosphorus, followed by colorimetric analysis by long-pathlength spectrophotometry (UV-1800 Shimadzu, Japan) using the molybdate blue method60, for which the method detection limit was ~ 0.005 mg P L−1. Between 2009 and 2016, soil samples (0–25 cm depth) were collected each autumn from all plots for determination of soil test P (STP) by the Bray-1 method61, using as an extractant a dilute hydrochloric acid and ammonium fluoride solution, as is recommended for neutral to slightly acidic soils. The measured STP concentration in mg P kg−1 was converted to kg P ha−1 based on soil sampling depth and soil bulk density (mean, 1.5 g cm−3). Sampling of water samples from lakes, streams and wells for P determination In addition to chemistry of soil and subsurface soil water in the BCSE, waters from lakes, streams, and residential water supply wells were also sampled during 2009–2016 for TDP analysis using Supor 450 membrane filters and the same analytical method as for soil water. These water bodies are within 15 km of the study site, within a landscape mosaic of row crops, grasslands, deciduous forest, and wetlands, with some residential development (Supplementary Fig. S2, Supplementary Table S2). Details of land use and cover change in the vicinity of KBS are given in Hamilton et al.48, and patterns in nutrient concentrations in local surface waters are further discussed in Hamilton62. Leaching estimates, modeled drainage, and data analysis Leaching was estimated at daily time steps and summarized as total leaching on a crop-year basis, defined from the date of planting or leaf emergence in a given year to the day prior to planting or emergence in the following year. TDP concentrations (mg L−1) of subsurface soil water were linearly interpolated between sampling dates during non-freezing periods (April–November) and over non-sampling periods (December–March) based on the preceding November and subsequent April samples. Daily rates of TDP leaching (kg ha−1) were calculated by multiplying concentration (mg L−1) by drainage rates (m3 ha−1 day−1) modeled by the Systems Approach for Land Use Sustainability (SALUS) model, a crop growth model that is well calibrated for KBS soil and environmental conditions. SALUS simulates yield and environmental outcomes in response to weather, soil, management (planting dates, plant population, irrigation, N fertilizer application, and tillage), and genetics63. The SALUS water balance sub-model simulates surface runoff, saturated and unsaturated water flow, drainage, root water uptake, and evapotranspiration during growing and non-growing seasons63. The SALUS model has been used in studies of evapotranspiration48,51,64 and nutrient leaching20,65,66,67 from KBS soils, and its predictions of growing-season evapotranspiration are consistent with independent measurements based on growing-season soil water drawdown53 and evapotranspiration measured by eddy covariance68. Phosphorus leaching was assumed insignificant on days when SALUS predicted no drainage. Volume-weighted mean TDP concentrations in leachate for each crop-year and for the entire 7-year study period were calculated as the total dissolved P leaching flux (kg ha−1) divided by the total drainage (m3 ha−1). One-way ANOVA with time (crop-year) as the fixed factor was conducted to compare total annual drainage rates, P leaching rates, volume-weighted mean TDP concentrations, and maximum aboveground biomass among the cropping systems over all seven crop-years as well as with TDP concentrations from local lakes, streams, and groundwater wells. When a significant (α = 0.05) difference was detected among the groups, we used the Tukey honest significant difference (HSD) post-hoc test to make pairwise comparisons among the groups. In the case of maximum aboveground biomass, we used the Tukey–Kramer method to make pairwise comparisons among the groups because the absence of poplar data after the 2013 harvest resulted in unequal sample sizes. We also used the Tukey–Kramer method to compare the frequency distributions of TDP concentrations in all of the soil leachate samples with concentrations in lakes, streams, and groundwater wells, since each sample category had very different numbers of measurements.
Other
Individual spreadsheets in “data table_leaching_dissolved organic carbon and nitrogen.xls” 1. annual precip_drainage 2. biomass_corn, perennial grasses 3. biomass_poplar 4. annual N leaching _vol-wtd conc 5. Summary_N leached 6. annual DOC leachin_vol-wtd conc 7. growing season length 8. correlation_nh4 VS no3 9. correlations_don VS no3_doc VS don Each spreadsheet is described below along with an explanation of variates. Note that ‘nan’ indicate data are missing or not available. First row indicates header; second row indicates units 1. Spreadsheet: annual precip_drainage Description: Precipitation measured from nearby Kellogg Biological Station (KBS) Long Term Ecological Research (LTER) Weather station, over 2009-2016 study period. Data shown in Figure 1; original data source for precipitation (https://lter.kbs.msu.edu/datatables/7). Drainage estimated from SALUS crop model. Note that drainage is percolation out of the root zone (0-125 cm). Annual precipitation and drainage values shown here are calculated for growing and non-growing crop periods. Variate Description year year of the observation crop “corn” “switchgrass” “miscanthus” “nativegrass” “restored prairie” “poplar” precip_G precipitation during growing period (milliMeter) precip_NG precipitation during non-growing period (milliMeter) drainage_G drainage during growing period (milliMeter) drainage_NG drainage during non-growing period (milliMeter) 2. Spreadsheet: biomass_corn, perennial grasses Description: Maximum aboveground biomass measurements from corn, switchgrass, miscanthus, native grass and restored prairie plots in Great Lakes Bioenergy Research Center (GLBRC) Biomass Cropping System Experiment (BCSE) during 2009-2015. Data shown in Figure 2. Variate Description year year of the observation date day of the observation (mm/dd/yyyy) crop “corn” “switchgrass” “miscanthus” “nativegrass” “restored prairie” “poplar” replicate each crop has four replicated plots, R1, R2, R3 and R4 station stations (S1, S2 and S3) of samplings within the plot. For more details, refer to link (https://data.sustainability.glbrc.org/protocols/156) species plant species that are rooted within the quadrat during the time of maximum biomass harvest. See protocol for more information, refer to link (http://lter.kbs.msu.edu/datatables/36) For maize biomass, grain and whole biomass reported in the paper (weed biomass or surface litter are excluded). Surface litter biomass not included in any crops; weed biomass not included in switchgrass and miscanthus, but included in grass mixture and prairie. fraction Fraction of biomass biomass_plot biomass per plot on dry-weight basis (Grams_Per_SquareMeter) biomass_ha biomass (megaGrams_Per_Hectare) by multiplying column biomass per plot with 0.01 3. Spreadsheet: biomass_poplar Description: Maximum aboveground biomass measurements from poplar plots in Great Lakes Bioenergy Research Center (GLBRC) Biomass Cropping System Experiment (BCSE) during 2009-2015. Data shown in Figure 2. Note that poplar biomass was estimated from crop growth curves until the poplar was harvested in the winter of 2013-14. Variate Description year year of the observation method methods of poplar biomass sampling date day of the observation (mm/dd/yyyy) replicate each crop has four replicated plots, R1, R2, R3 and R4 diameter_at_ground poplar diameter (milliMeter) at the ground diameter_at_15cm poplar diameter (milliMeter) at 15 cm height biomass_tree biomass per plot (Grams_Per_Tree) biomass_ha biomass (megaGrams_Per_Hectare) by multiplying biomass per tree with 0.01 4. Spreadsheet: annual N leaching_vol-wtd conc Description: Annual leaching rate (kiloGrams_N_Per_Hectare) and volume-weighted mean N concentrations (milliGrams_N_Per_Liter) of nitrate (no3) and dissolved organic nitrogen (don) in the leachate samples collected from corn, switchgrass, miscanthus, native grass, restored prairie and poplar plots in Great Lakes Bioenergy Research Center (GLBRC) Biomass Cropping System Experiment (BCSE) during 2009-2016. Data for nitrogen leached and volume-wtd mean N concentration shown in Figure 3a and Figure 3b, respectively. Note that ammonium (nh4) concentration were much lower and often undetectable (<0.07 milliGrams_N_Per_Liter). Also note that in 2009 and 2010 crop-years, data from some replicates are missing. Variate Description crop “corn” “switchgrass” “miscanthus” “nativegrass” “restored prairie” “poplar” crop-year year of the observation replicate each crop has four replicated plots, R1, R2, R3 and R4 no3 leached annual leaching rates of nitrate (kiloGrams_N_Per_Hectare) don leached annual leaching rates of don (kiloGrams_N_Per_Hectare) vol-wtd no3 conc. Volume-weighted mean no3 concentration (milliGrams_N_Per_Liter) vol-wtd don conc. Volume-weighted mean don concentration (milliGrams_N_Per_Liter) 5. Spreadsheet: summary_N leached Description: Summary of total amount and forms of N leached (kiloGrams_N_Per_Hectare) and the percent of applied N lost to leaching over the seven years for corn, switchgrass, miscanthus, native grass, restored prairie and poplar plots in Great Lakes Bioenergy Research Center (GLBRC) Biomass Cropping System Experiment (BCSE) during 2009-2016. Data for nitrogen amount leached shown in Figure 4a and percent of applied N lost shown in Figure 4b. Note the fraction of unleached N includes in harvest, accumulation in root biomass, soil organic matter or gaseous N emissions were not measured in the study. Variate Description crop “corn” “switchgrass” “miscanthus” “nativegrass” “restored prairie” “poplar” no3 leached annual leaching rates of nitrate (kiloGrams_N_Per_Hectare) don leached annual leaching rates of don (kiloGrams_N_Per_Hectare) N unleached N unleached (kiloGrams_N_Per_Hectare) in other sources are not studied % of N applied N lost to leaching % of N applied N lost to leaching 6. Spreadsheet: annual DOC leachin_vol-wtd conc Description: Annual leaching rate (kiloGrams_Per_Hectare) and volume-weighted mean N concentrations (milliGrams_Per_Liter) of dissolved organic carbon (DOC) in the leachate samples collected from corn, switchgrass, miscanthus, native grass, restored prairie and poplar plots in Great Lakes Bioenergy Research Center (GLBRC) Biomass Cropping System Experiment (BCSE) during 2009-2016. Data for DOC leached and volume-wtd mean DOC concentration shown in Figure 5a and Figure 5b, respectively. Note that in 2009 and 2010 crop-years, water samples were not available for DOC measurements. Variate Description crop “corn” “switchgrass” “miscanthus” “nativegrass” “restored prairie” “poplar” crop-year year of the observation replicate each crop has four replicated plots, R1, R2, R3 and R4 doc leached annual leaching rates of nitrate (kiloGrams_Per_Hectare) vol-wtd doc conc. volume-weighted mean doc concentration (milliGrams_Per_Liter) 7. Spreadsheet: growing season length Description: Growing season length (days) of corn, switchgrass, miscanthus, native grass, restored prairie and poplar plots in the Great Lakes Bioenergy Research Center (GLBRC) Biomass Cropping System Experiment (BCSE) during 2009-2015. Date shown in Figure S2. Note that growing season is from the date of planting or emergence to the date of harvest (or leaf senescence in case of poplar). Variate Description crop “corn” “switchgrass” “miscanthus” “nativegrass” “restored prairie” “poplar” year year of the observation growing season length growing season length (days) 8. Spreadsheet: correlation_nh4 VS no3 Description: Correlation of ammonium (nh4+) and nitrate (no3-) concentrations (milliGrams_N_Per_Liter) in the leachate samples from corn, switchgrass, miscanthus, native grass, restored prairie and poplar plots in Great Lakes Bioenergy Research Center (GLBRC) Biomass Cropping System Experiment (BCSE) during 2013-2015. Data shown in Figure S3. Note that nh4+ concentration in the leachates was very low compared to no3- and don concentration and often undetectable in three crop-years (2013-2015) when measurements are available. Variate Description crop “corn” “switchgrass” “miscanthus” “nativegrass” “restored prairie” “poplar” date date of the observation (mm/dd/yyyy) replicate each crop has four replicated plots, R1, R2, R3 and R4 nh4 conc nh4 concentration (milliGrams_N_Per_Liter) no3 conc no3 concentration (milliGrams_N_Per_Liter) 9. Spreadsheet: correlations_don VS no3_doc VS don Description: Correlations of don and nitrate concentrations (milliGrams_N_Per_Liter); and doc (milliGrams_Per_Liter) and don concentrations (milliGrams_N_Per_Liter) in the leachate samples of corn, switchgrass, miscanthus, native grass, restored prairie and poplar plots in Great Lakes Bioenergy Research Center (GLBRC) Biomass Cropping System Experiment (BCSE) during 2013-2015. Data of correlation of don and nitrate concentrations shown in Figure S4 a and doc and don concentrations shown in Figure S4 b. Variate Description crop “corn” “switchgrass” “miscanthus” “nativegrass” “restored prairie” “poplar” year year of the observation don don concentration (milliGrams_N_Per_Liter) no3 no3 concentration (milliGrams_N_Per_Liter) doc doc concentration (milliGrams_Per_Liter) More>>
The Abundance Matching Box for the Epoch of Reionization (AMBER) is a semi-numerical code for modeling the cosmic dawn. The new algorithm is not based on the excursion set formalism for reionization, but takes the novel approach of calculating the reionization-redshift field${z}_{\mathrm{re}}(\mathit{x})$assuming that hydrogen gas encountering higher radiation intensity are photoionized earlier. Redshift values are assigned while matching the abundance of ionized mass according to a given mass-weighted ionization fraction${\overline{x}}_{\mathrm{i}}(z)$. The code has the unique advantage of allowing users to directly specify the reionization history through the redshift midpointz_{mid}, duration Δ_{z}, and asymmetryA_{z}input parameters. The reionization process is further controlled through the minimum halo massM_{min}for galaxy formation and the radiation mean free pathl_{mfp}for radiative transfer. We implement improved methods for constructing density, velocity, halo, and radiation fields, which are essential components for modeling reionization observables. We compare AMBER with two other semi-numerical methods and find that our code more accurately reproduces the results from radiation-hydrodynamic simulations. The parallelized code is over four orders of magnitude faster than radiative transfer simulations and will efficiently enable large-volume models, full-sky mock observations, and parameter-space studies. AMBER will be mademore »publicly available to facilitate and transform studies of the Epoch of Reionization.
Bringmann, Kathrin, Kaszian, Jonas, Milas, Antun, and Nazaroglu, Caner. Integral representations of rank two false theta functions and their modularity properties. Retrieved from https://par.nsf.gov/biblio/10350001. Research in the Mathematical Sciences 8.4 Web. doi:10.1007/s40687-021-00284-1.
Bringmann, Kathrin, Kaszian, Jonas, Milas, Antun, & Nazaroglu, Caner. Integral representations of rank two false theta functions and their modularity properties. Research in the Mathematical Sciences, 8 (4). Retrieved from https://par.nsf.gov/biblio/10350001. https://doi.org/10.1007/s40687-021-00284-1
Bringmann, Kathrin, Kaszian, Jonas, Milas, Antun, and Nazaroglu, Caner.
"Integral representations of rank two false theta functions and their modularity properties". Research in the Mathematical Sciences 8 (4). Country unknown/Code not available. https://doi.org/10.1007/s40687-021-00284-1.https://par.nsf.gov/biblio/10350001.
@article{osti_10350001,
place = {Country unknown/Code not available},
title = {Integral representations of rank two false theta functions and their modularity properties},
url = {https://par.nsf.gov/biblio/10350001},
DOI = {10.1007/s40687-021-00284-1},
abstractNote = {Abstract False theta functions form a family of functions with intriguing modular properties and connections to mock modular forms. In this paper, we take the first step towards investigating modular transformations of higher rank false theta functions, following the example of higher depth mock modular forms. In particular, we prove that under quite general conditions, a rank two false theta function is determined in terms of iterated, holomorphic, Eichler-type integrals. This provides a new method for examining their modular properties and we apply it in a variety of situations where rank two false theta functions arise. We first consider generic parafermion characters of vertex algebras of type $$A_2$$ A 2 and $$B_2$$ B 2 . This requires a fairly non-trivial analysis of Fourier coefficients of meromorphic Jacobi forms of negative index, which is of independent interest. Then we discuss modularity of rank two false theta functions coming from superconformal Schur indices. Lastly, we analyze $${\hat{Z}}$$ Z ^ -invariants of Gukov, Pei, Putrov, and Vafa for certain plumbing $$\mathtt{H}$$ H -graphs. Along the way, our method clarifies previous results on depth two quantum modularity.},
journal = {Research in the Mathematical Sciences},
volume = {8},
number = {4},
author = {Bringmann, Kathrin and Kaszian, Jonas and Milas, Antun and Nazaroglu, Caner},
}