A recovery of Brouncker's proof for the quadrature continued fraction

350 years ago in Spring of 1655 Sir William Brouncker on a request by John Wallis obtained a beautiful continued fraction for 4/π. Brouncker never published his proof. Many sources on the history of Mathematics claim that this proof was lost forever. In this paper we recover the original proof from Wallis' remarks presented in his "Arithmetica Infinitorum". We show that Brouncker's and Wallis' formulas can be extended to MacLaurin's sinusoidal spirals via related Euler's products. We derive Ramanujan's formula from Euler's formula and, by using it, then show that numerators of convergents of Brouncker's continued fractions coincide up to a rotation with Wilson's orthogonal polynomials corresponding to the parameters a = 0, b = 1/2, c = d = 1/4

A Singular Riesz Product in the Nevai Class and Inner Functions with the Schur Parameters in ∩p>2lp

AbstractThere exist singular Riesz products dσ=∏∞κ=1(1+Re(ακζnκ)) on the unit circle T with the parameters (an)n⩾0 of orthogonal polynomials in L2(dσ) satisfying ∑∞n=0|an|p<+∞ for every p, p>2. The Schur parameters of the inner factor of the Cauchy integral ∫T(ζ−z)−1dσ(ζ), σ being such a Riesz product, belong to ∩p>2lp

Towards the assignment for the 41S04 ^1S_0 meson nonet

The strong decays of the $\pi(2070)$, $\eta(2010)$, $\eta(2100)$, $\eta(2190)$, and $\eta(2225)$ as the $4 ^1S_0$ quark-antiquark states are investigated in the framework of the $^3P_0$ meson decay model. It is found that the $\pi(2070)$, $\eta(2100)$, and $\eta(2225)$ appear to be the convincing $4 ^1S_0$ $q\bar{q}$ states while the assignment of the $\eta(2010)$ and $\eta(2190)$ as the $4 ^1S_0$ isoscalar states is not favored by their widths. In the presence of the $\pi(2070)$, $\eta(2100)$, and $\eta(2225)$ being the members of the $4 ^1S_0$ meson nonet, the $4 ^1S_0$ kaon is phenomenologically determined to has a mass of about 2153 MeV. The width of this unobserved kaon is expected to be about 197 MeV in the $^3P_0$ decay model.Comment: 15 pages, 5 figures, version accepted for publication in Physical Review

The abstract inverse scattering problem and the instability of completeness of orthogonal systems

AbstractUsing the instability of completeness of orthogonal systems we prove that every contractive operator-valued function S(t),t ∈ T, on a Hilbert space E is the scattering operator of a pair (U, Ů) of unitary operators on L2(E), where Ů is the shift Ůƒ = z · ƒ. A generalization of Weyl's criterion for an operator not to be essentially left invertible is also proved. We apply the result obtained to the general theory of orthogonal systems, to the construction of complete minimal families which are not hereditarily complete, and to the scattering theory

Review Essay: Where Will Presidential Autocracy Take Russia?

This book describes the Russian presidency between 1990 and 1996, its society, and its politics in the “Second Russian Republic” between 1993 and 1996. Exam- ining one of the most dramatic periods in the history of Russia, Nichols begins with Mikhail Gorbachev’s attempt to pull down the old authoritarian system and to push Russia onto a democratic path of development

General and complete disarmament is a guarantee of peace and security for all nations: Speech at the World Congress for General Disarmament and Peace, delivered July 10, 1962

https://stars.library.ucf.edu/prism/1529/thumbnail.jp

Szeg\H{o}'s theorem for matrix orthogonal polynomials

We extend some classical theorems in the theory of orthogonal polynomials on the unit circle to the matrix case. In particular, we prove a matrix analogue of Szeg\H{o}'s theorem. As a by-product, we also obtain an elementary proof of the distance formula by Helson and Lowdenslager.Comment: 19 pp, typos corrected, introduction expande

Dynamics of thermal annealing of fiber gratings directly written by an infrared femtosecond laser

The inscription of fibre Bragg gratings using infrared femtosecond laser offers a number of advantages over conventional methods based on UV inscription. The refractive index modification in femtosecond inscription is independent on defect formation and therefore should not experiment the defect-related thermal decay of UV inscribed gratings. In this paper, the response to thermal annealing of fiber Bragg gratings inscribed using a tightly focussed femtosecond laser is investigated. Experimental results reveal a vastly improve thermal stability compared to gratings inscribed using conventional methods based on UV light. Erasure was not observed until temperatures in the range between 900°C and 1000°. These devices are therefore particularly suited to work in hostile environments and as high temperature sensors