Factorization in SLn(R)SL_n(R) with elementary matrices when RR is the disk algebra and the Wiener algebra

Let $R$ be the polydisc algebra or the Wiener algebra. It is shown that the group $SL_n(R)$ is generated by the subgroup of elementary matrices with all diagonal entries $1$ and at most one nonzero off-diagonal entry. The result an easy consequence of the deep result due to Ivarsson and Kutzschebauch (Ann. of Math. 2012).Comment: 5 page

Projective freeness of algebras of real symmetric functions

Let D^n be the closed unit polydisk in C^n. Consider the ring C_r of complex-valued continuous functions on D^n that are real symmetric, that is, f(z)=(f(z^*))^* for all z in D^n. It is shown that C_r is projective free, that is, finitely generated projective modules over C_r are free. We also show that several subalgebras of the real symmetric polydisc algebra are projective free.Comment: 14 page

A summation method based on the Fourier series of periodic distributions and an example

A generalised summation method is considered based on the Fourier series of periodic distributions. It is shown that $$e^{it}-2e^{2it}+3e^{3it}-4e^{4it}+-\cdots = {\mathrm P\mathrm f} {\displaystyle \frac{e^{it}}{(1+e^{it})^2}} +i\pi \displaystyle \sum_{n\in \mathbb{Z}} \delta'_{(2n+1)\pi},$$ where ${\mathrm P\mathrm f} {\displaystyle \frac{e^{it}}{(1+e^{it})^2}}\in \mathcal{D}'(\mathbb{R})$ is the $2\pi$-periodic distribution given by \begin{eqnarray*} \left\langle {\mathrm P\mathrm f} {\displaystyle \frac{e^{it}}{(1+e^{it})^2}} ,\varphi \right\rangle &=& \lim_{\epsilon\searrow 0} \left( \int_{(-\delta,\pi-\epsilon)\cup(\pi+\epsilon,2\pi+\delta)}\frac{\varphi(t) e^{it}}{(1+e^{it})^2}dt -\frac{\varphi(\pi)}{\tan (\epsilon/2)}\right) \end{eqnarray*} $\varphi \in \mathcal{D}(\mathbb{R})$ with support $\textrm{supp}(\varphi)\subset (-\delta,2\pi+\delta)$, where $\delta\in (0,\pi)$. Applying the generalised summation method, we determine the sum of the divergent series $1+2+3+\cdots$, and more generally $1^k+2^k+3^k+\cdots$ for $k\in \mathbb{N}$.Comment: 31 pages, 5 figures (Typos corrected.

Noncoherence of the multiplier algebra of the Drury-Arveson space H^2_n for n>=3

Let H_n^2 denote the Drury-Arveson Hilbert space on the unit ball B_n in C^n, and let M(H_n^2) be its multiplier algebra. We show that for n>=3, the ring M(H_n^2) is not coherent.Comment: 10 page

Neutrino mass and dark energy constraints from redshift-space distortions

Cosmology in the near future promises a measurement of the sum of neutrino masses, a fundamental Standard Model parameter, as well as substantially-improved constraints on the dark energy. We use the shape of the BOSS redshift-space galaxy power spectrum, in combination with CMB and supernova data, to constrain the neutrino masses and the dark energy. Essential to this calculation are several recent advances in non-linear cosmological perturbation theory, including FFT methods, redshift space distortions, and scale-dependent growth. Our 95% confidence upper bound of 180 meV on the sum of masses degrades substantially to 540 meV when the dark energy equation of state and its first derivative are also allowed to vary, representing a significant challenge to current constraints. We also study the impact of additional galaxy bias parameters, finding that a greater allowed range of scale-dependent bias only slightly shifts the preferred neutrino mass value, weakens its upper bound by about 20%, and has a negligible effect on the other cosmological parameters.Comment: Matches accepted version. Code available at github.com/upadhye/redTim

Shannon's sampling theorem in a distributional setting

The classical Shannon sampling theorem states that a signal f with Fourier transform F in L^2(R) having its support contained in (-\pi,\pi) can be recovered from the sequence of samples (f(n))_{n in Z} via f(t)=\sum_{n in Z} f(n) (sin(\pi (t -n)))/(\pi (t-n)) (t in R). In this article we prove a generalization of this result under the assumption that F is a compactly supported distribution with its support contained in (-\pi,\pi).Comment: This paper has been withdrawn by the author due to an error in a claim about singular supports in the proo

On Isosceles Triangles and Related Problems in a Convex Polygon

Given any convex $n$-gon, in this article, we: (i) prove that its vertices can form at most $n^2/2 + \Theta(n\log n)$ isosceles trianges with two sides of unit length and show that this bound is optimal in the first order, (ii) conjecture that its vertices can form at most $3n^2/4 + o(n^2)$ isosceles triangles and prove this conjecture for a special group of convex $n$-gons, (iii) prove that its vertices can form at most $\lfloor n/k \rfloor$ regular $k$-gons for any integer $k\ge 4$ and that this bound is optimal, and (iv) provide a short proof that the sum of all the distances between its vertices is at least $(n-1)/2$ and at most $\lfloor n/2 \rfloor \lceil n/2 \rceil(1/2)$ as long as the convex $n$-gon has unit perimeter

Parallel implementation of three-dimensional molecular dynamic simulation for laser-cluster interaction

The objective of this article is to report the parallel implementation of the 3D molecular dynamic simulation code for laser-cluster interactions. The benchmarking of the code has been done by comparing the simulation results with some of the experiments reported in the literature. Scaling laws for the computational time is established by varying the number of processor cores and number of macroparticles used. The capabilities of the code are highlighted by implementing various diagnostic tools. To study the dynamics of the laser-cluster interactions, the executable version of the code is available from the author.Comment: 5 pages, 7 figure

Texture zeroes and discrete flavor symmetries in light and heavy Majorana neutrino mass matrices: a bottom-up approach

Texture zeroes in neutrino mass matrix $M_\nu$ may give us hints about the symmetries involved in neutrino mass generation. We examine the viability of such texture zeroes in a model independent way through a bottom-up approach. Using constraints from the neutrino oscillation data, we develop an analytic framework that can identify these symmetries and quantify deviations from them. We analyze the textures of $M_\nu$ as well as those of $M_M$, the mass matrix of heavy Majorana neutrinos in the context of Type-I seesaw. We point out how the viability of textures depends on the absolute neutrino mass scale, the neutrino mass ordering and the mixing angle $\theta_{13}$. We also examine the compatibility of discrete flavor symmetries like $\mu$--$\tau$ exchange and $S_3$ permutation with the current data. We show that the $\mu-\tau$ exchange symmetry for $M_\nu$ can be satisfied for any value of the absolute neutrino mass, but for $M_\nu$ to satisfy the $S_3$ symmetry, neutrino masses have to be quasi-degenerate. On the other hand, both these symmetries are currently allowed for $M_M$ for all values of absolute neutrino mass and both mass orderings.Comment: 33 pages, minor revision in section-VI (A

On two natural extensions of Vinnicombe's metric: their noncoincidence yet equivalence on stabilizable plants over A_+

Let A_+ be the ring of Laplace transforms of complex Borel measures on R with support in [0,+\infty) which do not have a singular nonatomic part. We compare the nu-metric d_{A_+} for stabilizable plants over A_+ given in the article by Ball and Sasane [2010], with yet another metric d_{H^\infty}|_{A_+}, namely the one induced by the metric d_{H^\infty} for the set of stabilizable plants over H^\infty given in teh article by Sasane in 2011. Both d_{A_+} and d_{H^\infty} coincide with the classical Vinnicombe metric defined for rational transfer functions, but we show here by means of an example that these two possible extensions of the classical nu-metric for plants over A_+ do not coincide on the set of stabilizable plants over A_+. We also prove that they nevertheless give rise to the same topology on stabilizable plants over A_+, which in turn coincides with the gap metric topology.Comment: 13 pages, 0 figure