On spectral sets of integers

Based on tiles and on the Coven-Meyerowitz property, we present some examples and some general constructions of spectral subsets of integers

Early Results from VLT-SPHERE: Long-Slit Spectroscopy of 2MASS 0122-2439B, a Young Companion Near the Deuterium Burning Limit

We present 0.95-1.80 $\mu$m spectroscopy of the $\sim$12-27 $M_{\rm Jup}$ companion orbiting the faint ($R$$\sim$13.6), young ($\sim$120 Myr) M-dwarf 2MASS J01225093--2439505 ("2M0122--2439 B") at 1.5 arcsecond separation (50 AU). Our coronagraphic long-slit spectroscopy was obtained with the new high contrast imaging platform VLT-SPHERE during Science Verification. The unique long-slit capability of SPHERE enables spectral resolution an order of magnitude higher than other extreme AO exoplanet imaging instruments. With a low mass, cool temperature, and very red colors, 2M0122-2439 B occupies a particularly important region of the substellar color-magnitude diagram by bridging the warm directly imaged hot planets with late-M/early-L spectral types (e.g. $\beta$ Pic b and ROXs 42Bb) and the cooler, dusty objects near the L/T transition (e.g. HR 8799bcde and 2MASS 1207b). We fit BT-Settl atmospheric models to our $R$$\approx$350 spectrum and find $T_{\rm eff}$=1600$\pm$100 K and $\log(g)$=4.5$\pm$0.5 dex. Visual analysis of our 2M0122-2439 B spectrum suggests a spectral type L3-L4, and we resolve shallow $J$-band alkali lines, confirming its low gravity and youth. Specifically, we use the Allers & Liu (2013) spectral indices to quantitatively measure the strength of the FeH, VO, KI, spectral features, as well as the overall $H$-band shape. Using these indices, along with the visual spectral type analysis, we classify 2M0122-2439 B as an intermediate gravity (INT-G) object with spectral type L3.7$\pm$1.0.Comment: Accepted to ApJ Letters, 8 pages, 4 figures, some minor typographical issues were fixe

Polar Smectic Films

We report on a new experimental procedure for forming and studying polar smectic liquid crystal films. A free standing smectic film is put in contact with a liquid drop, so that the film has one liquid crystal/liquid interface and one liquid crystal/air interface. This polar environment results in changes in the textures observed in the film, including a boojum texture and a previously unobserved spiral texture in which the winding direction of the spiral reverses at a finite radius from its center. Some aspects of these textures are explained by the presence of a Ksb term in the bulk elastic free energy density that favors a combination of splay and bend deformations.Comment: 4 pages, REVTeX, 3 figures, submitted to PR

Sums of random polynomials with differing degrees

Let $\mu$ and $\nu$ be probability measures in the complex plane, and let $p$ and $q$ be independent random polynomials of degree $n$, whose roots are chosen independently from $\mu$ and $\nu$, respectively. Under assumptions on the measures $\mu$ and $\nu$, the limiting distribution for the zeros of the sum $p+q$ was by computed by Reddy and the third author [J. Math. Anal. Appl. 495 (2021) 124719] as $n \to \infty$. In this paper, we generalize and extend this result to the case where $p$ and $q$ have different degrees. In this case, the logarithmic potential of the limiting distribution is given by the pointwise maximum of the logarithmic potentials of $\mu$ and $\nu$, scaled by the limiting ratio of the degrees of $p$ and $q$. Additionally, our approach provides a complete description of the limiting distribution for the zeros of $p + q$ for any pair of measures $\mu$ and $\nu$, with different limiting behavior shown in the case when at least one of the measures fails to have a logarithmic moment.Comment: 30 pages, 2 figures. Major update containing new methods and improved result

On the Real Roots and Real Eigenvalues of the Generalized Large Box Model for Random Polynomials and Random Matrices

Let n and N be in N, and for 0 &le; i &le; n&minus;1, let &alpha;i &lt; &beta;i &isin; R. Consider the monic polynomial in a single complex variable of the form fn(z) = zn + an&minus;1zn&minus;1 +&middot;&middot;&middot; +a1z +a0 whose coefficients ai are uniformly distributed on [&alpha;iN,&beta;iN]&cap;Z for each 0 &le; i &le; n&minus;1 and jointly independent. This random polynomial model is referred to as the generalized large box model. When instead &alpha;ij &lt; &beta;ij for 1 &le; i,j &le; n and one considers the n-by-n random matrix whose entries are uniformly distributed on [&alpha;ijN,&beta;ijN] &cap; Z for each 1 &le; i,j &le; n and jointly independent, we say that the matrix is drawn from the generalized large box model ensemble. This thesis is organized into five chapters. Chapter 1 develops and presents the history of the relevant random polynomial and random matrix models, related results, and notation. Chapters 2 and 3 are concerned with finding the probability that random polynomials whose coefficients obey the generalized large box model have all real roots, as N &rarr; &infin;. Specifically, in Chapter 2, discriminant and root analysis methods are applied to low degree polynomials, obtaining explicit answers. These methods further find an extremely dominant root, denoted by &xi;n, for all degrees; this is a root whose modulus is not tight as N &rarr; &infin;, while the moduli of the remaining roots are tight as N &rarr; &infin;. This expands upon on a discovery made by Dubickas and Sha [Exp. Math., 24(3):312&ndash;325, 2015]. As N &rarr; &infin;, we show that &xi;n is real with probability tending to 1 and that |&xi;n + an&minus;1| converges in distribution to |X/Y |, where X is uniformly distributed on [&alpha;n&minus;2,&beta;n&minus;2], Y is uniformly distributed on [&alpha;n&minus;1,&beta;n&minus;1], and X and Y are independent. In Chapter 3, we consider non-monic degree n&minus;1 polynomials whose coefficients are uniformly distributed on [&alpha;i,&beta;i] for 0 &le; i &le; n &minus; 1 and jointly independent, referred to as generalized bounded height model polynomials. As N &rarr; &infin;, the probability that the degree n generalized large box model polynomial with coefficients uniformly distributed on [&alpha;iN,&beta;iN] &cap; Z for 0 &le; i &le; n&minus;1 and jointly independent has all real roots converges to the probability that the degree n &minus; 1 generalized bounded height model polynomial with coefficients uniformly distributed on [&alpha;i,&beta;i] for 0 &le;i&le;n&minus;1andjointly independent has all real roots. The methods of Bert&acute;ok, Hajdu, and Peth&uml;o [J. Number Theory, 179:172&ndash;184, 2017] are used to express this probability in terms of an integral formula. For the special case when &alpha;i = 0 and &beta;i = 1 for 0 &le; i &le; n&minus;1, a relation to the Selberg integral is explored and we show that the probability of such a polynomial having all real roots is positive and monotonically decreasing in n. In Chapters 4 and 5, the analogous question of the probability that the random matrix whose entries are drawn from the generalized large box model ensemble has all real eigenvalues is considered. In Chapter 4 we begin by letting &alpha;ij &lt; &beta;ij for 1 &le; i,j &le; n and consider the n-by n random matrix whose entries are uniformly distributed on [&alpha;ij,&beta;ij] for each 1 &le; i,j &le; n and jointly independent. This matrix ensemble is referred to as the generalized bounded height ensemble. Using Edelman&rsquo;s method [J. Multivariate Anal., 60(2):203&ndash;232, 1997], we factor these matrices into their real Schur decomposition and present an integral formula for the probability that generalized bounded height ensemble matrices have all real roots. In Chapter 5, we show that if A is an n-by-n random matrix with entries that are uniformly distributed on [&alpha;ijN,&beta;ijN] &cap; Z for 1 &le; i,j &le; n and jointly independent and B is an n-by-n random matrix with entries that are uniformly distributed on [&alpha;ij,&beta;ij] for 1 &le; i,j &le; n and jointly independent, then for each 0 &le; k &le; n, as N &rarr;&infin;,theprobability that A has exactly k real eigenvalues converges to the probability that B has exactly k real eigenvalues. Moreover, the empirical spectral measure of A/N converges weakly in distribution to the empirical spectral measure of B and the joint distribution of the eigenvalues of A/N converges in distribution to the joint distribution of the eigenvalues of B. Finally, still in Chapter 5, we consider rank one perturbations of the random matrix A whose entries are all independently and identically (iid) uniformly distributed on [&minus;N,N]&cap;Z. We say that Ais drawn from the large box model ensemble. Letting P be the matrix whose entries are all &micro;N, we consider the limiting spectral behavior of A+P in the three cases where &micro;N/N &rarr; &infin;,&micro;N/N &rarr; 0, and &micro;N/N &rarr; c, for c &isin; R. If limN&rarr;&infin; &micro;N N =&infin;, the eigenvalues of A+P &micro;N converge almost surely to the eigenvalues of P/&micro;N, and the largest eigenvalue (in magnitude) is real with probability tending to 1. Additionally, the centered and correctly normalized largest eigenvalue of A + P converges in distribution to a Bates distribution with n2 parameters. If limN&rarr;&infin; &micro;N N =0, the empirical spectral measures of A+P N and A N both converge weakly in distribution to the empirical spectral measure of the matrix whose entries are iid and uniformly distributed on [&minus;1,1]; the joint distribution of the eigenvalues of A N and A+P N both converge in distribution to the joint distribution of eigenvalues of the same random matrix. If limN&rarr;&infin; &micro;N N =c&gt;0,the empirical spectral measure of A+P N converges weakly in distribution to the empirical spectral measure of the matrix whose entries are iid and uniformly distributed on [&minus;1 + c,1 + c]. In addition, if c &gt; 12, we show that as N &rarr; &infin;, A+P N has exactly one real outlier eigenvalue and provide a range for its location.</p

Scaling of Spectra of Cantor-Type Measures and Some Number Theoretic Considerations

We investigate some relations between number theory and spectral measures related to the harmonic analysis of a Cantor set. Specifically, we explore ways to determine when an odd natural number m generates a complete or incomplete Fourier basis for a Cantor-type measure with scale g

A liquid crystalline copper phthalocyanine derivative for high performance organic thin film transistors

This journal is © The Royal Society of Chemistry 2012Bottom-gate, bottom-contact organic thin film transistors (OTFTs) were fabricated using solvent soluble copper 1,4,8,11,15,18,22,25-octakis(hexyl)phthalocyanine as the active semiconductor layer. The compound was deposited as 70 nm thick spin-coated films onto gold source–drain electrodes supported on octadecyltrichlorosilane treated 250 nm thick SiO2 gate insulators. The performance of the OTFTs was optimised by investigating the effects of vacuum annealing of the films at temperatures between 50 0C and 200 0C, a range that included the thermotropic mesophase of the bulk material. These effects were monitored by ultraviolet-visible absorption spectroscopy, atomic force microscopy and XRD measurements. Device performance was shown to be dependent upon the annealing temperature due to structural changes of the film. Devices heat treated at 100 0C under vacuum (≥10-7 mbar) were found to exhibit the highest field-effect mobility, 0.7 cm2 V^-1 s^-1, with an on–off current modulation ratio of~107, a reduced threshold voltage of 2.0 V and a sub-threshold swing of 1.11 V per decade.UK Technology Strategy Board (Project no: TP/6/EPH/6/S/K2536J) and UK National Measurement System (Project IRD C02 ‘‘Plastic Electronics’’, 2008–2011)