Discrete Mathematics

The purpose of the present work is to provide short and supple teaching notes for a $30$ hours introductory course on elementary \textit{Enumerative Algebraic Combinatorics}. We fully adopt the \textit{Rota way} (see, e.g. \cite{KY}). The themes are organized into a suitable sequence that allows us to derive any result from the preceding ones by elementary processes. Definitions of \textit{combinatorial coefficients} are just by their \textit{combinatorial meaning}. The derivation techniques of formulae/results are founded upon constructions and two general and elementary principles/methods: - The \textit{bad element} method (for \textit{recursive} formulae). As the reader should recognize, the bad element method might be regarded as a combinatorial companion of the idea of \textit{conditional probability}. - The \textit{overcounting} principle (for \textit{close form} formulae). Therefore, \textit{no computation} is required in \textit{proofs}: \textit{computation formulae are byproducts of combinatorial constructions}. We tried to provide a self-contained presentation: the only prerequisite is standard high school mathematics. We limited ourselves to the \textit{combinatorial point of view}: we invite the reader to draw the (obvious) \textit{probabilistic interpretations}

Quantum immanants, double Young-Capelli bitableaux and Schur shifted symmetric functions

We propose a new method for a unified study of some of the main features of the theory of the center $\boldsymbol{\zeta}(n)$ of the enveloping algebra U(gl(n)) and of the algebra $\Lambda^*(n)$ of shifted symmetric polynomials, that allows the whole theory to be developed, in a transparent and concise way, from the representation-theoretic point of view, that is entirely in the center of U(gl(n)). Our methodological innovation is the systematic use of the superalgebraic method of virtual variables for gl(n), which is, in turn, an extension of Capelli's method of ``variabili ausiliarie''. The passage $n \rightarrow \infty$ for the algebras $\boldsymbol{\zeta}(n)$ and $\Lambda^*(n)$ is here obtained both as direct and inverse limit in the category of filtered algebras. The present approach leads to proofs that are almost direct consequences of the definitions and constructions: they often reduce to a few lines computation

Plume and surface feature structure and compositional effects on Europa's global exosphere: Preliminary Europa mission predictions

AbstractA Europa plume source, if present, may produce a global exosphere with complex spatial structure and temporal variability in its density and composition. To investigate this interaction we have integrated a water plume source containing multiple organic and nitrile species into a Europan Monte Carlo exosphere model, considering the effect of Europa's gravity in returning plume ejecta to the surface, and the subsequent spreading of adsorbed and exospheric material by thermal desorption and re-sputtering across the entire body. We consider sputtered, radiolytic and potential plume sources, together with surface adsorption, regolith diffusion, polar cold trapping, and re-sputtering of adsorbed materials, and examine the spatial distribution and temporal evolution of the exospheric density and composition. These models provide a predictive basis for telescopic observations (e.g. HST, JWST) and planned missions to the Jovian system by NASA and ESA. We apply spacecraft trajectories to our model to explore possible exospheric compositions which may be encountered along proposed flybys of Europa to inform the spatial and temporal relationship of spacecraft measurements to surface and plume source compositions. For the present preliminary study, we have considered four cases: Case A: an equatorial flyby through a sputtered only exosphere (no plumes), Case B: a flyby over a localized sputtered ‘macula’ terrain enriched in non-ice species, Case C: a south polar plume with an Enceladus-like composition, equatorial flyby, and Case D: a south polar plume, flyby directly through the plume

Detection of negative pickup ions at Saturn's moon Dione

We investigate a possible negative ion feature observed by the Cassini Plasma Spectrometer (CAPS) during a flyby of Saturn's moon Dione that occurred on April 7, 2010. By examining possible particle trajectories, we find that the observed particles are consistent with negative pick‐up ions originating near the moon's surface. We find that the mass of the negative pick‐up ions is in the range of 15 – 25 u and tentatively identify this species as O‐, likely resulting from ionization and subsequent pick‐up from Dione's O2‐CO2 exosphere. Our estimates show that the negative ion density is ~3 x 10‐3 cm‐3. This is comparable to, but slightly smaller than, that previously reported for the density of O2+ pick‐up ions for the same flyby, indicating that negative pick‐up ions may represent a major loss channel for Dione's exosphere

Parameter Estimation from Time-Series Data with Correlated Errors: A Wavelet-Based Method and its Application to Transit Light Curves

We consider the problem of fitting a parametric model to time-series data that are afflicted by correlated noise. The noise is represented by a sum of two stationary Gaussian processes: one that is uncorrelated in time, and another that has a power spectral density varying as $1/f^\gamma$. We present an accurate and fast [O(N)] algorithm for parameter estimation based on computing the likelihood in a wavelet basis. The method is illustrated and tested using simulated time-series photometry of exoplanetary transits, with particular attention to estimating the midtransit time. We compare our method to two other methods that have been used in the literature, the time-averaging method and the residual-permutation method. For noise processes that obey our assumptions, the algorithm presented here gives more accurate results for midtransit times and truer estimates of their uncertainties.Comment: Accepted in ApJ. Illustrative code may be found at http://www.mit.edu/~carterja/code/ . 17 page

Multiscale periodicities in aerosol optical depth over India

Aerosols exhibit periodic or cyclic variations depending on natural and anthropogenic sources over a region, which can get modulated by synoptic meteorological parameters such as winds, rainfall and relative humidity, and long-range transport. Information on periodicity and phase in aerosol properties assumes significance in prediction as well as to examine the radiative and climate effects of aerosols including its association with changes in cloud properties and rainfall. Periodicity in aerosol optical depth, which is a columnar measure of aerosol distribution, is determined using continuous wavelet transform over 35 locations in India. Continuous wavelet transform is used in the study because it is better suited to extract the periodic and local modulations present at various frequency ranges, as these features are invisible in conventional methods such as Fourier Transform. Monthly mean aerosol optical depths (AODs) from MODerate Resolution Imaging Spectroradiometer (MODIS) on board the Terra satellite from January 2001 to December 2012 are used. Annual and quasi-biennial oscillations (QBO) in AOD are evident in addition to the weak semi-annual and quasi-triennial. The semi-annual and annual oscillations are consistent with the seasonal and yearly cycle of variations in AODs. The 40-month periodicity indicates the presence of long term correlations in AOD. The observed periodicities in MODIS Terra AODs are also evident in the ground-based AOD measurements made over Kanpur in the Indo-Gangetic Plain. The phase of the periodicity in AOD is stable in the mid-frequency range, while local disturbances in the high-frequency range and long term changes in the atmospheric composition give rise to unstable phases in low-frequency range. That modulations in AOD over one location/region can influence the other is revealed by the presence of phase relation among different locations

Compaction of microporous amorphous solid water by ion irradiation

We have studied the compaction of vapor-deposited amorphous solid water by energetic ions at 40 K. The porosity was characterized by ultraviolet-visible spectroscopy, infrared spectroscopy, and methane adsorption/desorption. These three techniques provide different and complementary views of the structural changes in ice resulting from irradiation. We find that the decrease in internal surface area of the pores, signaled by infrared absorption by dangling bonds, precedes the decrease in the pore volume during irradiation. Our results imply that impacts from cosmic rays can cause compaction in the icy mantles of the interstellar grains, which can explain the absence of dangling bond features in the infrared spectrum of molecular clouds.Fil: Raut, U.. University of Virginia; Estados UnidosFil: Teolis, B. D.. University of Virginia; Estados UnidosFil: Loeffler, M. J.. University of Virginia; Estados UnidosFil: Vidal, Ricardo Alberto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Desarrollo Tecnológico para la Industria Química. Universidad Nacional del Litoral. Instituto de Desarrollo Tecnológico para la Industria Química; ArgentinaFil: Famá, M.. University of Virginia; Estados UnidosFil: Baragiola, R. A.. University of Virginia; Estados Unido

The composition and structure of the Enceladus plume

The Cassini Ultraviolet Imaging Spectrograph (UVIS) observed an occultation of the Sun by the water vapor plume at the south polar region of Saturn\u27s moon Enceladus. The Extreme Ultraviolet (EUV) spectrum is dominated by the spectral signature of H(2)O gas, with a nominal line-of-sight column density of 0.90 +/- 0.23 x 10(16) cm(-2) (upper limit of 1.0 x 10(16) cm(-2)). The upper limit for N(2) is 5 x 10(13) cm(-2), or \u3c 0.5% in the plume; the lack of N(2) has significant implications for models of the geochemistry in Enceladus\u27 interior. The inferred rate of water vapor injection into Saturn\u27s magnetosphere is similar to 200 kg/s. The calculated values of H(2)O flux from three occultations observed by UVIS have a standard deviation of 30 kg/s (15%), providing no evidence for substantial short-term variability. Collimated gas jets are detected in the plume with Mach numbers of 5-8, implying vertical gas velocities that exceed 1000 m/sec. Observations at higher altitudes with the Cassini Ion Neutral Mass Spectrometer indicate correlated structure in the plume. Our results support the subsurface liquid model, with gas escaping and being accelerated through nozzle-like channels to the surface, and are consistent with recent particle composition results from the Cassini Cosmic Dust Analyzer