On a conjecture of Wilf

Let n and k be natural numbers and let S(n,k) denote the Stirling numbers of the second kind. It is a conjecture of Wilf that the alternating sum \sum_{j=0}^{n} (-1)^{j} S(n,j) is nonzero for all n>2. We prove this conjecture for all n not congruent to 2 and not congruent to 2944838 modulo 3145728 and discuss applications of this result to graph theory, multiplicative partition functions, and the irrationality of p-adic series.Comment: 18 pages, final version, accepted for publication in the Journal of Combinatorial Theory, Series

Highly eccentric inspirals into a black hole

We model the inspiral of a compact stellar-mass object into a massive nonrotating black hole including all dissipative and conservative first-order-in-the-mass-ratio effects on the orbital motion. The techniques we develop allow inspirals with initial eccentricities as high as $e\sim0.8$ and initial separations as large as $p\sim 50$ to be evolved through many thousands of orbits up to the onset of the plunge into the black hole. The inspiral is computed using an osculating elements scheme driven by a hybridized self-force model, which combines Lorenz-gauge self-force results with highly accurate flux data from a Regge-Wheeler-Zerilli code. The high accuracy of our hybrid self-force model allows the orbital phase of the inspirals to be tracked to within $\sim0.1$ radians or better. The difference between self-force models and inspirals computed in the radiative approximation is quantified.Comment: Updated to reflect published versio

Evolution of small-mass-ratio binaries with a spinning secondary

We calculate the evolution and gravitational-wave emission of a spinning compact object inspiraling into a substantially more massive (non-rotating) black hole. We extend our previous model for a non-spinning binary [Phys. Rev. D 93, 064024] to include the Mathisson-Papapetrou-Dixon spin-curvature force. For spin-aligned binaries we calculate the dephasing of the inspiral and associated waveforms relative to models that do not include spin-curvature effects. We find this dephasing can be either positive or negative depending on the initial separation of the binary. For binaries in which the spin and orbital angular momentum are not parallel, the orbital plane precesses and we use a more general osculating element prescription to compute inspirals.Comment: 17 pages, 6 figure

Fast spectral source integration in black hole perturbation calculations

This paper presents a new technique for achieving spectral accuracy and fast computational performance in a class of black hole perturbation and gravitational self-force calculations involving extreme mass ratios and generic orbits. Called \emph{spectral source integration} (SSI), this method should see widespread future use in problems that entail (i) point-particle description of the small compact object, (ii) frequency domain decomposition, and (iii) use of the background eccentric geodesic motion. Frequency domain approaches are widely used in both perturbation theory flux-balance calculations and in local gravitational self-force calculations. Recent self-force calculations in Lorenz gauge, using the frequency domain and method of extended homogeneous solutions, have been able to accurately reach eccentricities as high as $e \simeq 0.7$. We show here SSI successfully applied to Lorenz gauge. In a double precision Lorenz gauge code, SSI enhances the accuracy of results and makes a factor of three improvement in the overall speed. The primary initial application of SSI--for us its \emph{raison d'\^{e}tre}--is in an arbitrary precision \emph{Mathematica} code that computes perturbations of eccentric orbits in the Regge-Wheeler gauge to extraordinarily high accuracy (e.g., 200 decimal places). These high accuracy eccentric orbit calculations would not be possible without the exponential convergence of SSI. We believe the method will extend to work for inspirals on Kerr, and will be the subject of a later publication. SSI borrows concepts from discrete-time signal processing and is used to calculate the mode normalization coefficients in perturbation theory via sums over modest numbers of points around an orbit. A variant of the idea is used to obtain spectral accuracy in solution of the geodesic orbital motion.Comment: 15 pages, 7 figure

Extreme-mass-ratio inspirals into a black hole

The inspiral of a compact stellar-mass object (black hole, neutron star, or white dwarf) into a super-massive black hole is modeled perturbatively. All dissipative and conservative effects on the orbital motion are included up to first-order in the mass-ratio via the gravitational self-force. To calculate the self-force, a number of advanced numerical tools were developed. These tools are able to describe the complete range of astrophysically relevant orbital configurations for the first time (for non-rotating binary components), and at an accuracy never before achieved. The accuracy is high enough to aid in the detection of gravitational waves while correctly determining the physical parameters of the source.Doctor of Philosoph

Repeated faint quasinormal bursts in extreme-mass-ratio inspiral waveforms: Evidence from frequency-domain scalar self-force calculations on generic Kerr orbits

We report development of a code to calculate the scalar self-force on a scalar-charged particle moving on generic bound orbits in the Kerr spacetime. The scalar self-force model allows rapid development of computational techniques relevant to generic gravitational extreme-mass-ratio inspirals (EMRIs). Our frequency-domain calculations are made with arbitrary numerical precision code written in \textsc{Mathematica}. We extend spectral source integration techniques to the Kerr spacetime, increasing computational efficiency. We model orbits with nearly arbitrary inclinations $0\leq\iota<\pi/2$ and eccentricities up to $e \lesssim 0.8$. This effort extends earlier work by Warburton and Barack where motion was restricted to the equatorial plane or to inclined spherical orbits. Consistent with a recent discovery by Thornburg and Wardell \cite{ThorWard17} in time-domain calculations, we observe self-force oscillations during the radially-outbound portion of highly eccentric orbits around a rapidly rotating black hole. As noted previously, these oscillations reflect coupling into the self-force by quasinormal modes excited during pericenter passage. Our results confirm the effect with a frequency-domain code. \emph{More importantly, we find that quasinormal bursts (QNBs) appear directly in the waveform following each periastron passage.} These faint bursts are shown to be a superposition of the least-damped overtone (i.e., fundamental) of at least four ($l=m \le 4$) quasinormal modes. Our results suggest that QNBs should appear in gravitational waveforms, and thus provide a gauge-invariant signal. Potentially observable in high signal-to-noise ratio EMRIs, QNBs would provide high-frequency components to the parameter estimation problem that would complement low-frequency elements of the waveform.Comment: 28 pages, 11 figures, 5 tables; Updated to reflect published versio

Chromophoric Dissolved Organic Matter and Dissolved Organic Carbon from Sea-Viewing Wide Field-of-View Sensor (SeaWiFS), Moderate Resolution Imaging Spectroradiometer (MODIS) and MERIS Sensors: Case Study for the Northern Gulf of Mexico

Empirical band ratio algorithms for the estimation of colored dissolved organic matter (CDOM) and dissolved organic carbon (DOC) for Sea-viewing Wide Field-of-view Sensor (SeaWiFS), Moderate Resolution Imaging Spectroradiometer (MODIS) and MERIS ocean color sensors were assessed and developed for the northern Gulf of Mexico. Match-ups between in situ measurements of CDOM absorption coefficients at 412 nm (aCDOM(412)) with that derived from SeaWiFS were examined using two previously reported reflectance band ratio algorithms. Results indicate better performance using the Rrs(510)/Rrs(555) (Bias = −0.045; RMSE = 0.23; SI = 0.49, and R2 = 0.66) than the Rrs(490)/Rrs(555) reflectance band ratio algorithm. Further, a comparison of aCDOM(412) retrievals using the Rrs(488)/Rrs(555) for MODIS and Rrs(510)/Rrs(560) for MERIS reflectance band ratios revealed better CDOM retrievals with MERIS data. Since DOC cannot be measured directly by remote sensors, CDOM as the colored component of DOC is utilized as a proxy to estimate DOC remotely. A seasonal relationship between CDOM and DOC was established for the summer and spring-winter with high correlation for both periods (R2~0.9). Seasonal band ratio empirical algorithms to estimate DOC were thus developed using the relationships between CDOM-Rrs and seasonal CDOM-DOC for SeaWiFS, MODIS and MERIS. Results of match-up comparisons revealed DOC estimates by both MODIS and MERIS to be relatively more accurate during summer time, while both of them underestimated DOC during spring-winter time. A better DOC estimate from MERIS in comparison to MODIS in spring-winter could be attributed to its similarity with the SeaWiFS band ratio CDOM algorithm

Junior Recital: Connor Osburn, horn

This recital is presented in partial fulfillment of requirements for the degree Bachelor of Music in Performance. Mr. Osburn studies horn with Tom Witte.https://digitalcommons.kennesaw.edu/musicprograms/1306/thumbnail.jp