Ramanujan's "Lost Notebook" and the Virasoro Algebra

By using the theory of vertex operator algebras, we gave a new proof of the famous Ramanujan's modulus 5 modular equation from his "Lost Notebook" (p.139 in \cite{R}). Furthermore, we obtained an infinite list of $q$-identities for all odd moduli; thus, we generalized the result of Ramanujan.Comment: To appear in Comm. Math. Phy

Tshekedi Khama papers : covering dates: 1889 - 1997

This collection comprises 10 linear metres of the papers of Tshekedi Khama (1905-1959), Regent of the Bangwato Tribe, and uncle of Seretse Khama. The collection also includes some papers of close family members

Spartan Daily February 25, 2010

Volume 134, Issue 15https://scholarworks.sjsu.edu/spartandaily/1231/thumbnail.jp

Determination of new coefficients in the angular momentum and energy fluxes at infinity to 9PN for eccentric Schwarzschild extreme-mass-ratio inspirals using mode-by-mode fitting

We present an extension of work in an earlier paper showing high precision comparisons between black hole perturbation theory and post-Newtonian (PN) theory in their region of overlapping validity for bound, eccentric-orbit, Schwarzschild extreme-mass-ratio inspirals. As before we apply a numerical fitting scheme to extract eccentricity coefficients in the PN expansion of the gravitational wave fluxes, which are then converted to exact analytic form using an integer-relation algorithm. In this work, however, we fit to individual $lmn$ modes to exploit simplifying factorizations that lie therein. Since the previous paper focused solely on the energy flux, here we concentrate initially on analyzing the angular momentum flux to infinity. A first step involves finding convenient forms for hereditary contributions to the flux at low-PN order, analogous to similar terms worked out previously for the energy flux. We then apply the upgraded techniques to find new PN terms through 9PN order and (at many PN orders) to $e^{30}$ in the power series in eccentricity. With the new approach applied to angular momentum fluxes, we return to the energy fluxes at infinity to extend those previous results. Like before, the underlying method uses a \textsc{Mathematica} code based on use of the Mano-Suzuki-Takasugi (MST) function expansion formalism to represent gravitational perturbations and spectral source integration (SSI) to find numerical results at arbitrarily high precision.Comment: 36 pages, 1 figur

Optimal binomial, Poisson, and normal left-tail domination for sums of nonnegative random variables

Let $X_1,\dots,X_n$ be independent nonnegative random variables (r.v.'s), with $S_n:=X_1+\dots+X_n$ and finite values of $s_i:=E X_i^2$ and $m_i:=E X_i>0$. Exact upper bounds on $E f(S_n)$ for all functions $f$ in a certain class $\mathcal{F}$ of nonincreasing functions are obtained, in each of the following settings: (i) $n,m_1,\dots,m_n,s_1,\dots,s_n$ are fixed; (ii) $n$, $m:=m_1+\dots+m_n$, and $s:=s_1+\dots+s_n$ are fixed; (iii)~only $m$ and $s$ are fixed. These upper bounds are of the form $E f(\eta)$ for a certain r.v. $\eta$. The r.v. $\eta$ and the class $\mathcal{F}$ depend on the choice of one of the three settings. In particular, $(m/s)\eta$ has the binomial distribution with parameters $n$ and $p:=m^2/(ns)$ in setting (ii) and the Poisson distribution with parameter $\lambda:=m^2/s$ in setting (iii). One can also let $\eta$ have the normal distribution with mean $m$ and variance $s$ in any of these three settings. In each of the settings, the class $\mathcal{F}$ contains, and is much wider than, the class of all decreasing exponential functions. As corollaries of these results, optimal in a certain sense upper bounds on the left-tail probabilities $P(S_n\le x)$ are presented, for any real $x$. In fact, more general settings than the ones described above are considered. Exact upper bounds on the exponential moments $E\exp\{hS_n\}$ for $h<0$, as well as the corresponding exponential bounds on the left-tail probabilities, were previously obtained by Pinelis and Utev. It is shown that the new bounds on the tails are substantially better.Comment: Version 2: fixed a typo (p. 17, line 2) and added a detail (p. 17, line 9). Version 3: Added another proof of Lemma 3.2, using the Redlog package of the computer algebra system Reduce (open-source and freely distributed

The Faculty Notebook, May 2005

The Faculty Notebook is published periodically by the Office of the Provost at Gettysburg College to bring to the attention of the campus community accomplishments and activities of academic interest. Faculty are encouraged to submit materials for consideration for publication to the Associate Provost for Faculty Development. Copies of this publication are available at the Office of the Provost

Spartan Daily, October 12, 2006

Volume 127, Issue 27https://scholarworks.sjsu.edu/spartandaily/10285/thumbnail.jp