On affinity relating two positive measures and the connection coefficients between polynomials orthogonalized by these measures

We consider two positive, normalized measures dA(x) and dB(x) related by the relationship dA(x)=(C/(x+D))dB(x) or by dA(x) = (C/(x^2+E))dB(x) and dB(x) is symmetric. We show that then the polynomial sequences {a_{n}(x)}, {b_{n}(x)} orthogonal with respect to these measures are related by the relationship a_{n}(x)=b_{n}(x)+{\kappa}_{n}b_{n-1}(x) or by a_{n}(x) = b_{n}(x) + {\lambda}_{n}b_{n-2}(x) for some sequences {{\kappa}_{n}} and {{\lambda}_{n}}. We present several examples illustrating this fact and also present some attempts for extensions and generalizations. We also give some universal identities involving polynomials {b_{n}(x)} and the sequence {{\kappa}_{n}} that have a form of Fourier series expansion of the Radon--Nikodym derivative of one measure with respect to the other

A few remarks on orthogonal polynomials

Knowing a sequence of moments of a given, infinitely supported, distribution we obtain quickly: coefficients of the power series expansion of monic polynomials $\left\{ p_{n}\right\} _{n\geq 0}$ that are orthogonal with respect to this distribution, coefficients of expansion of $x^{n}$ in the series of $p_{j},$ $j\leq n$, two sequences of coefficients of the 3-term recurrence of the family of $\left\{ p_{n}\right\} _{n\geq 0}$, the so called "linearization coefficients" i.e. coefficients of expansion of $$ in the series of $p_{j},$ $j\leq m+n.$\newline Conversely, assuming knowledge of the two sequences of coefficients of the 3-term recurrence of a given family of orthogonal polynomials $\left\{ p_{n}\right\} _{n\geq 0},$ we express with their help: coefficients of the power series expansion of $p_{n}$, coefficients of expansion of $x^{n}$ in the series of $p_{j},$ $j\leq n,$ moments of the distribution that makes polynomials $\left\{ p_{n}\right\} _{n\geq 0}$ orthogonal. \newline Further having two different families of orthogonal polynomials $\left\{ p_{n}\right\} _{n\geq 0}$ and $\left\{ q_{n}\right\} _{n\geq 0}$ and knowing for each of them sequences of the 3-term recurrences, we give sequence of the so called "connection coefficients" between these two families of polynomials. That is coefficients of the expansions of $p_{n}$ in the series of $q_{j},$ $j\leq n.$\newline We are able to do all this due to special approach in which we treat vector of orthogonal polynomials $\left\{ p_{j}\left( x)\right) \right\} _{j=0}^{n}$ as a linear transformation of the vector $\left\{ x^{j}\right\} _{j=0}^{n}$ by some lower triangular $(n+1)\times (n+1)$ matrix $\mathbf{\Pi }_{n}.$Comment: 18 page

On the generalized Kesten--McKay distributions

We examine the properties of distributions with the density of the form: $\frac{2A_{n}c^{n-2}\sqrt{c^{2}-x^{2}}}{\pi \prod_{j=1}^{n}(c(1+a_{j}^{2})-2a_{j}x)},$ where $c,a_{1},\ldots ,a_{n}$ are some parameters and $A_{n}$ a suitable constant. We find general forms of $A_{n}$, of $k-$th moment and of $k-$th polynomial orthogonal with respect to such measures. We also calculate Cauchy transforms of these measures. We indicate connections of such distributions with distributions and polynomials forming the so-called Askey--Wilson scheme. On the way, we prove several identities concerning rational symmetric functions. Finally, we consider the case of parameters $a_{1},\ldots ,a_{n}$ forming conjugate pairs and give some multivariate interpretations based on the obtained distributions at least for the cases $n=2,4,6.$Comment: 14 page

The acquisition of sentence alternations : how children understand and use the English dative alternation.

Many English verbs expressing transfer can be used in two different constructions, one with no preposition (Rick gave Kate a coffee) and one with the preposition to (Rick gave a coffee to Kate). Whenever speakers use such a verb, they have to choose between these two constructions. This choice is determined in part by some features of the two objects: all other things being equal, speakers are more likely to use whichever construction places a shorter object before a longer one (Rick gave a coffee to the tall and well-dressed woman standing next the the desk at the southern side of the room), an animate object before an inanimate one (Rick gave Kate a coffee), a plural object before a singular one (Rick gave Kate and Roy an espresso machine), and so on. This system of feature-based choices is established very well for adult language using language corpora and experiments, but there are fewer corpora and experimental studies of child language. Because of this dearth of data, it is unknown how children acquire this choice-making system: do they start making choices determined by only one of these features and add the others piecemeal, or do they learn the system wholesale and only tweak which features win out over others? The three experiments in this thesis are a first step in answering this question. They are designed to map out the effects of length, animacy, and grammatical number on these choices over the course of typical first language acquisition. Because animacy is less stable a concept than length and number, the first experiment measures children’s and adults’ conceptions of animacy more indirectly. The second experiment presents the same participants with sentences using give where one of the two objects has been replaced by noise, and measures which of a constrained set of options they gaze at and which they choose to fill the noise gap. This provides measures of their expectations and preferences for the length, animacy, and number of the objects in these gaps. The third experiment has participants reproduce give sentences with different combinations of animacy, number, and construction. Participants reproduce sentences that conform to their choice-making system more easily. The results of these three experiments show that children as young as four years already prefer the animate-before-inanimate order. The shorter-before-longer preference is not found in any age group when the difference in lengths is just one syllable. This evidence adds to a growing body of literature converging on the finding that choices in ordering phenomena are affected by the same features wherever these phenomena occur, throughout language acquisition as well as across languages. Data from the second experiment also substantiates the common assumption that touchscreen input and eye gaze are both closely linked to attention. This will allow researchers in the cognitive sciences to use touchscreens as an alternative to eyetracking more confidently

A modeling study of ion implantation in crystalline silicon involving Monte Carlo and molecular dynamics methods

Ph.DDOCTOR OF PHILOSOPH

Electron, Photon, and Positron Scattering Dynamics of Complex Molecular Targets

Electron scattering cross sections have been computed for pyridine and pyrimidine using the static-exchange approximation with model potential to account for dynamic electron correlation. To obtain well-converged orbitals, we have expanded all partial waves to a maximum angular momentum of l = 60 for both targets. We have obtained total cross sections for electron scattering energies to 20 eV. Both targets display similar features, namely a dipole-induced increase in the integrated cross section at scattering energies below 5 eV, and peaks corresponding to resonances in b1, a2, and b1 symmetries. These resonances were investigated through a Siegert eigenstate analysis and Breit-Wigner fit of the SECP eigenphase sums. They were also compared to the virtual orbitals obtained from a minimum basis set Hartree-Fock calculation on both targets. We consider electron scattering resonances from cis-diamminedichloroplatinum, [Pt(NH3)2Cl2], the ligand molecular species Cl2 (1Sigma+g ), and the isolated transition metal center Pt in a nondegenerate atomic state (1S) at the SECP level of theory. As a rigorous comparison to the single-state, single-configuration SECP level results of these smaller, yet electron dense targets, we have also considered scattering from ground state Cl2 and Pt in the 1S and 3D states in the multichannel configuration-interaction (MCCI) approximation originally developed for photoionization for scattering up to 10 eV. Photoionization cross sections and angular distributions in the recoil frame (RFPAD) and molecular frame (MFPAD) have been computed for inner-shell C 1s and Cl 2p ionization from the chloroalkanes chloromethane and chloroethane, with ionization leading to a variety of ionic fragment states. We have also computed valence level ionization from the nitro molecule nitromethane CH3NO2 leading to the dissociation of the CN bond. All of these calculations were performed in the frozen-core Hartree-Fock approximation. Even at this level of theory, we obtain computed results that compare well to the photoelectronphotoion coincidence measurements. The fullerene C20 is the smallest fullerene predicted to exist, with most relevant structural calculations suggesting the reduction of the icosahedral symmetry into one in which the target species possesses at maximum only a dihedral axis. We have computed positron scattering cross sections for the molecule in two low-symmetry structural isomers Ci and C2, within the HF approximation. Density functional expressions were used to incorporate important positron-electron interactions within the calculation. We have found similar cross sections and resonance features for both isomers, including a positron scattering resonance whose density is found within the framework of the fullerene cluster