Zagier's weight 3/23/2 mock modular form

Mock modular forms have their origins in Ramanujan's pioneering work on mock theta functions. In a 1975 paper, Zagier proved certain transformation properties of the generating function of the Hurwitz class numbers $H(n)$ for the discriminant $(-n)$. In the modern framework, these results show that the generating function of $H(n)$ is a mock modular form of weight 3/2 with the theta function being the shadow. In this expository paper, we provide a detailed proof of Zagier's result.Comment: 24 page

An Analogue of Weil's Converse Theorem for Harmonic Maass Forms of Polynomial Growth

We construct a family of examples of harmonic Maass forms of polynomial growth for any level whose shadows are Eisenstein series of integral weight. We further consider Dirichlet series attached to a harmonic Maass form of polynomial growth, study its analytic properties, and prove an analogue of Weil's converse theorem.Comment: 28 Page

Beauty And The Beast Part 2: Apprehending The Missing Supercurrent

The Moonshine module is a $c=24$ conformal field theory (CFT) whose automorphism group is the Monster group. It was argued by Dixon, Ginsparg, and Harvey in \cite{Dixon:1988qd} that there exists a spin lift of the Moonshine CFT with superconformal symmetry. Reference \cite{Dixon:1988qd} did not provide an explicit construction of a superconformal current. The present paper fills that gap. In fact, we will construct several superconformal currents in a spin lift of the Moonshine CFT using techniques developed in \cite{Harvey:2020jvu}. In particular, our construction relies on error correcting codes.Comment: 66 page

Mock Modularity In CHL Models

Dabholkar, Murthy and Zagier (DMZ) proved that there is a canonical decomposition of a meromorphic Jacobi form of integral index for $\mathrm{SL}(2, \mathbb{Z})$ with poles on torsion points $z\in\mathbb{Q}\tau+\mathbb{Q}$ into polar and finite parts, and showed that the finite part is a mock Jacobi form. In this paper we generalize the results of DMZ to meromorphic Jacobi forms of rational index for congruence subgroups of $\mathrm{SL}(2, \mathbb{Z})$. As an application, we establish that a large class of single-centered black hole degeneracies in CHL models are given by the Fourier coefficients of mock Jacobi forms. In this process we refine the result of DMZ regarding the set of charges for which the single-centered black hole degeneracies are given by a mock modular form. In particular, in the case studied by DMZ, we present examples of charges for which the single-centered degeneracies are not captured by the mock modular form of the expected index.Comment: 64 Page

DESMIDS BIODIVERSITY IN FRESHWATER PONDS OF DISTRICT NAGAUR, WESTERN RAJASTHAN INDIA: A PRELIMINARY INVESTIGATION

The present study was carried out to investigate the desmids biodiversity in traditional freshwater reservoirs of Nagaur district, Western Rajasthan, India. The samples were collected in January 2021 from the freshwater small ponds which harvests rainwater located in and around Nagaur city. These samples were analyzed by following the standard method prescribed by A.P.H.A for assessing the various physicochemical parameters such as pH, TDS, Fluoride, Nitrate, Chloride, total alkalinity, and total hardness to check drinking water quality and for ecological assessment. Furthermore desmids species-level identification and description have been done with the help of standard literatures and monographs. Results of physicochemical parameters indicate the deteriorating drinking water quality of all four sampling sites. In the present study total of 16 species belong to 5 genera i.e. Netrium, Closterium, Cosmerium, Staurastrum, and Eustrum of desmids have been recorded for the first time from the freshwater ponds located in or around the Nagaur city. The higher biodiversity of Closterium and Cosmarium species at Jhada talab confirm its significant relationship with high calcium and nutrient-rich water quality. While low biodiversity of desmids at remaining all village ponds indicates that high pH and oligotrophic quality decreases the desmids biodiversity. Results of this study might be helpful to assess the geographic distribution of desmids flora of Nagaur district and establish water quality of protected freshwater habitats as well as in updating the list of current species of desmids in this sub-region

Screening of Bacterial Strains for Polygalacturonase Activity: Its Production by Bacillus sphaericus (MTCC 7542)

At present almost all the pectinolytic enzymes used for industrial applications are produced by fungi. There are a few reports of pectinase production by bacterial strains. Therefore, in the present study, seventy-four bacterial strains, isolated from soil and rotten vegetable samples, were screened for polygalacturonase production. The strain PG-31, which gave maximum activity, was identified as Bacillus sphaericus (MTCC 7542). Maximal quantities of polygalacturonase were produced when a 16-hours-old inoculum was used at 7.5% (v/v) in production medium and incubated in shaking conditions (160 rpm) for 72 hours. The optimal temperature and pH for bacterial growth and polygalacturonase production were found to be 30°C and 6.8, respectively. Maximum enzyme production resulted when citrus pectin was used as the carbon source at a concentration of 1.25% (w/v), whereas other carbon sources led to a decrease (30%–70%) in enzyme production. Casein hydrolysate and yeast extract used together as organic nitrogen source gave best results, and ammonium chloride was found to be the most suitable inorganic nitrogen source. The supplementation of media with 0.9% (w/v) D-galacturonic acid led to a 23% increase in activity. Bacillus sphaericus, a bacterium isolated from soil, produced good amount of polygalacturonase activity at neutral pH; hence, it would be potentially useful to increase the yield of banana, grape, or apple juice