Some vanishing sums involving binomial coefficients in the denominator

Identities involving binomial coeffcients usually arise in situations where counting is carried out in two different ways. For instance, some identities obtained by William Horrace [1] using probability theory turn out to be special cases of the Chu-Vandermonde identities. Here, we obtain some generalizations of the identities observed by Horrace and give different types of proofs; these, in turn, give rise to some other new identities. In particular, we evaluate sums of the form Pm j=0 (1) j j d (mj) (n+jj ) and deduce that they vanish when d is even and m = n > d=2. It is well-known [2] that sums involving binomial coeffcients can usually be expressed in terms of the hypergeometric functions but it is more interesting if such a function can be evaluated explicitly at a given argument. Identities such as the ones we prove could perhaps be of some interest due to the explicit evaluation possible. The papers [3], [4] are among many which deal with identities for sums where the binomial coeffcients occur in the denominator and we use similar methods here

Eigenforms of half-integral weight

Let k be an odd integer and N a positive integer such that 4 | N. Let X be a Dirichlet character modulo N. Shimura decomposes the space of half-integral weight forms Sk/2(N,X) as Sk/2(N,X) = S0(N,X)oOΦSk/2(N,X,Φ) where Φ runs through the newforms of weight k-1 and level dividing N/2 and character X2; Sk/2(N,X,Φ) is the subspace of forms that are Shimura-equivalent to Φ; and S0(N,X) is the subspace generated by single-variable theta-series. We give an explicit algorithm for computing this decomposition. Once we have the decomposition, we can exploreWaldspurger's theorem expressing the critical values of the L-functions of twists of an elliptic curve in terms of the coefficients of modular forms of half-integral weight. Following Tunnell, this often allows us to give a criterion for the n-th twist of an elliptic curve to have positive rank in terms of the number of representations of certain integers by certain ternary quadratic forms

Investigating The Impacts Of Ageing And Moisture On Dielectric Response Of Oil/Paper Insulation Systems

The oil/paper insulation system in a transformer degrades under various chemical, thermal and electromechanical stresses during its operation. Dielectric response measurements like the Return Voltage Measurement (RVM) and Polarisation and Depolarisation Current (PDC) measurement are currently being used for condition assessment of the insulation. In the current research project, the ageing condition and the moisture content of both paper and oil samples have been varied over a controlled range and their effects on the dielectric response measurements have been investigated. Experimental results are provided in this paper to demonstrate the effects of both ageing and moisture content of paper and oil on the RVM and PDC results

Effectiveness of light emitting diode versus conventional phototherapy in neonatal hyperbilirubinemia: a hospital based observational study

Background: The most commonly used light sources in the conventional phototherapy units are compact fluorescent lamp (CFL), halogen spotlights and fiberoptic blankets. Recently light emitting diodes (LED) has emerged as better light source for phototherapy and almost replacing all the available conventional light sources. Comparative studies on the efficacy of LED versus conventional phototherapy are limited from India. That is why; this study was undertaken.Methods: 48 neonates of ≥35 weeks gestational age with hyperbilirubinemia were participated in this study. Among them, 24 neonates received conventional phototherapy and rest of them received LED phototherapy. The rate of fall of bilirubin levels at 6 hours and at completion along with total duration of phototherapy in both groups was measured. Results analysed by standard statistical methods.Results: LED phototherapy units showed higher rate of fall in bilirubin at six hrs (LED group: 0.38±0.05 mg/dl/hr versus conventional group: 0.30±0.04 mg/dl/hr, p<0.05) and after completion of therapy (LED group:                      0.32±0.03 mg/dl/hr versus conventional group: 0.26±0.03 mg/dl/hr, p<0.05) compared to conventional group. Significant difference was documented in total duration of phototherapy in LED group (30.8±1.8 hours) when compared to conventional group (34.6±0.7 hours). None of the neonate showed phototherapy failure. Side effects were minimal and comparable in both the groups.Conclusions: The LED phototherapy units are more efficacious in terms of higher rate of fall of bilirubin levels and lesser duration of phototherapy compared to conventional phototherapy units

Identification of Medicinally Active Ingredient in Ultradiluted Digitalis purpurea: Fluorescence Spectroscopic and Cyclic-Voltammetric Study

Serially diluted and agitated (SAD) drugs available commercially are in use with great faith because of the astonishing results they produce. The scientific viewpoint attached to the centuries-old therapy with SAD drugs, as in homeopathy, remained doubtful for want of appropriate research and insufficient evidence base. The conflicting points related to SAD drug mostly related to the level of concentrations/dilutions, use of drug in contradictory clinical conditions compared to the modern system of medicine, identification of medicinally active ingredient in concentrations and dilutions used in commercially available SAD drugs, and lack of laboratory-based pharmacological data vis-à-vis modern medicine. Modus operandi of SAD drug is also unknown. To address some of these issues an analytical study was carried out wherein commercially available SAD drug Digitalis purpurea, commonly used in different systems of medicine, was put to test. Various concentrations of commercially available Digitalis purpurea were analyzed using analytical methods: cyclic voltammetry, emission spectroscopy, and UV-VIS spectroscopy. These analytical methods apparently identified the medicinal ingredients and effect of serial dilution in commercial preparation of the drugs