Biorthonormal systems, partial fractions, and Hermite interpolation

AbstractUsing some properties of dual bases in finite dimensional vector spaces we obtain elementary linear algebra proofs of the partial fractions decomposition and the Hermite interpolation theorems. We also obtain an explicit expression for the inverse of a confluent Vandermonde matrix, an algebraic version of the residue theorem for rational functions, and several inverse pairs of change of basis matrices on a Space of polynomials

Linearization and connection coefficients of polynomial sequences: A matrix approach

For a sequence of polynomials $\{p_k(t)\}$ in one real or complex variable, where $p_k$ has degree $k$, for $k\ge 0$, we find explicit expressions and recurrence relations for infinite matrices whose entries are the coefficients $d(n,m,k)$, called linearization coefficients, that satisfy $$p_n(t) p_m(t)=\sum_{k=0}^{n+m} d(n,m,k) p_k(t).$$ For any pair of polynomial sequences $\{u_k(t)\}$ and $\{p_k(t)\}$ we find infinite matrices whose entries are the coefficients $e(n,m,k)$ that satisfy $$p_n(t) p_m(t)=\sum_{k=0}^{n+m} e(n,m,k) u_k(t).$$ Such results are obtained using a matrix approach. We also obtain recurrence relations for the linearization coefficients, apply the general results to general orthogonal polynomial sequences and to particular families of orthogonal polynomials such as the Chebyshev, Hermite, and Charlier families.Comment: 14 page

The effect of stress on meiotic recombination in maize (Zea mays L)

Plant genomes have the capacity to change in response to abiotic stress and other environmental signals. In maize (Zea mays L.), enhanced genetic recombination in response to low temperature has been shown to alter recombination frequencies in maize (Shams-UI-Islam, 1956). The effect of other environmental stresses on genetic recombination has not been reported in maize. Therefore, the primary objective of this dissertation was to identify whether water-deficit stress during meiosis and defoliation during pre-meiosis will affect meiotic recombination in maize. A secondary objective was to observe crossover occurrence and distribution. Three experiments are presented in this dissertation. The first two experiments focus on the effect of water stress on meiotic recombination of two maize genotypes (B73/Mo17 and Mo17/H99). The third experiment investigates the effect of defoliation on meiotic recombination of Mo17/H99. For each experiment male meiotic recombination was observed in backcross populations derived from 3 stress and 3 non-stress plants. Progeny of each population was genotyped at microsatellite loci to create genetic maps for chromosomes 1 and 10. Comparisons of recombination were made within and between treatments. For water-deficit experiments genetic maps of chromosomes 1 and 10 were larger for stressed plants. Results suggest that increase on recombination under water-deficit stress is a general response in maize. Maps of populations subjected to defoliation did not differ in length from those of the control treatment

Cosmological implications of Primordial Black Holes

The possibility that a relevant fraction of the dark matter might be comprised of Primordial Black Holes (PBHs) has been seriously reconsidered after LIGO's detection of a $\sim 30 M_{\odot}$ binary black holes merger. Despite the strong interest in the model, there is a lack of studies on possible cosmological implications and effects on cosmological parameters inference. We investigate correlations with the other standard cosmological parameters using cosmic microwave background observations, finding significant degeneracies, especially with the tilt of the primordial power spectrum and the sound horizon at radiation drag. However, these degeneracies can be greatly reduced with the inclusion of small scale polarization data. We also explore if PBHs as dark matter in simple extensions of the standard $\Lambda$CDM cosmological model induces extra degeneracies, especially between the additional parameters and the PBH's ones. Finally, we present cosmic microwave background constraints on the fraction of dark matter in PBHs, not only for monochromatic PBH mass distributions but also for popular extended mass distributions. Our results show that extended mass distribution's constraints are tighter, but also that a considerable amount of constraining power comes from the high-$\ell$ polarization data. Moreover, we constrain the shape of such mass distributions in terms of the correspondent constraints on the PBH mass fraction.Comment: 20 pages, 9 figures. Matches version accepted to publish in JCA

Beware of commonly used approximations I: errors in forecasts

In the era of precision cosmology, establishing the correct magnitude of statistical errors in cosmological parameters is of crucial importance. However, widely used approximations in galaxy surveys analyses can lead to parameter uncertainties that are grossly mis-estimated, even in a regime where the theory is well understood (e.g., linear scales). These approximations can be introduced at three different levels: in the form of the likelihood, in the theoretical modelling of the observable and in the numerical computation of the observable. Their consequences are important both in data analysis through e.g., Markov Chain Monte Carlo parameter inference, and when survey instrument and strategy are designed and their constraining power on cosmological parameters is forecasted, for instance using Fisher matrix analyses. In this work, considering the galaxy angular power spectrum as the target observable, we report one example of approximation for each of such three categories: neglecting off-diagonal terms in the covariance matrix, neglecting cosmic magnification and using the Limber approximation on large scales. We show that these commonly used approximations affect the robustness of the analysis and lead, perhaps counter-intuitively, to unacceptably large mis-estimates of parameters errors (from few~$10\%$ up to few~$100\%$) and correlations. Furthermore, these approximations might even spoil the benefits of the nascent multi-tracer and multi-messenger cosmology. Hence we recommend that the type of analysis presented here should be repeated for every approximation adopted in survey design or data analysis, to quantify how it may affect the results. To this aim, we have developed \texttt{Multi\_CLASS}, a new extension of \texttt{CLASS} that includes the angular power spectrum for multiple (galaxy and other tracers such as gravitational waves) populations.Comment: 43 pages, 9 figures. Matches the published version. \texttt{Multi\_CLASS} is now available at https://github.com/nbellomo/Multi_CLAS

Multi-variate joint PDF for non-Gaussianities: exact formulation and generic approximations

We provide an exact expression for the multi-variate joint probability distribution function of non-Gaussian fields primordially arising from local transformations of a Gaussian field. This kind of non-Gaussianity is generated in many models of inflation. We apply our expression to the non- Gaussianity estimation from Cosmic Microwave Background maps and the halo mass function where we obtain analytical expressions. We also provide analytic approximations and their range of validity. For the Cosmic Microwave Background we give a fast way to compute the PDF which is valid up to 7{\sigma} for fNL values (both true and sampled) not ruled out by current observations, which consists of expressing the PDF as a combination of bispectrum and trispectrum of the temperature maps. The resulting expression is valid for any kind of non-Gaussianity and is not limited to the local type. The above results may serve as the basis for a fully Bayesian analysis of the non-Gaussianity parameter.Comment: Matches accepted verion to JCAP; conclusions unchaged, extra references adde

The bias of weighted dark matter halos from peak theory

We give an analytical form for the weighted correlation function of peaks in a Gaussian random field. In a cosmological context, this approach strictly describes the formation bias and is the main result here. Nevertheless, we show its validity and applicability to the evolved cosmological density field and halo field, using Gaussian random field realisations and dark matter N-body numerical simulations. Using this result from peak theory we compute the bias of peaks (and dark matter halos) and show that it reproduces results from the simulations at the ${\mathcal O}(10\%)$ level. Our analytical formula for the bias predicts a scale-dependent bias with two characteristics: a broad band shape which, however, is most affected by the choice of weighting scheme and evolution bias, and a more robust, narrow feature localised at the BAO scale, an effect that is confirmed in simulations. This scale-dependent bias smooths the BAO feature but, conveniently, does not move it. We provide a simple analytic formula to describe this effect. We envision that our analytic solution will be of use for galaxy surveys that exploit galaxy clustering.Comment: Submitted to MNRA