Nonlinear equations for the recurrence coefficients of discrete orthogonal polynomials

We study polynomials orthogonal on a uniform grid. We show that each weight function gives two potentials and each potential leads to a structure relation (lowering operator). These results are applied to derive second order difference equations satisfied by the orthogonal polynomials and nonlinear difference equations satisfied by the recursion coefficients in the three-term recurrence relations.     

q-Blossoming for analytic functions

Numerical Algorithms - We construct a q-analog of the blossom for analytic functions, the analytic q-blossom. This q-analog also extends the notion of q-blossoming from polynomials to analytic...     

Modified construction of nuclear Fréchet spaces without basis

Recently, B. Mitiagin and N. Zobin constructed an example of nuclear Fréchet space without basis. The essential modification of their constructions gives the following results. There exists such a nuclear Fréchet space X that for any nuclear Fréchet space Y the space X × Y has no basis (Sections 1 and 2). This fact has a lot of corollaries (Sect. 3); e.g., the space X × C^∞(R¹) having the maximal diametral dimension among nuclear Fréchet spaces nevertheless has no basis. One can also construct (Sect. 4) a nuclear Fréchet space $\text{X}\text{?}$ without strongly finite-dimensional decomposition (see Definition 0.1). In Section 5 some comments and open questions are given.     

Inverting the local geodesic X-ray transform on tensors

We prove the local invertibility, up to potential fields, and stability of the geodesic X-ray transform on tensor fields of order 1 and 2 near a strictly convex boundary point, on manifolds with boundary of dimension n ≥ 3. We also present an inversion formula. Under the condition that the manifold can be foliated with a continuous family of strictly convex surfaces, we prove a global result which also implies a lens rigidity result near such a metric. The class of manifolds satisfying the foliation condition includes manifolds with no focal points, and does not exclude existence of conjugate points.     

Nuclear limits on gravitational waves from elliptically deformed pulsars

Gravitational radiation is a fundamental prediction of General Relativity. Elliptically deformed pulsars are among the possible sources emitting gravitational waves (GWs) with a strain-amplitude dependent upon the star's quadrupole moment, rotational frequency, and distance from the detector. We show that the gravitational wave strain amplitude h₀$h_0$ depends strongly on the equation of state of neutron-rich stellar matter. Applying an equation of state with symmetry energy constrained by recent nuclear laboratory data, we set an upper limit on the strain-amplitude of GWs produced by elliptically deformed pulsars. Depending on details of the EOS, for several millisecond pulsars at distances 0.18 kpc to 0.35 kpc from Earth, the maximal  h₀$h_0$ is found to be in the range of ∼[0.4–1.5]×¹⁰⁻²⁴$\sim [0.4\text{–}1.5]\times {10}^{\mathrm{-24}}$. This prediction serves as the first direct nuclear constraint on the gravitational radiation. Its implications are discussed.     

Novel polynomial Bernstein bases and Bézier curves based on a general notion of polynomial blossoming

We introduce the G-blossom of a polynomial by altering the diagonal property of the classical blossom, replacing the identity function by arbitrary linear functions G=G(t). By invoking the G-blossom, we construct G-Bernstein bases and G-Bézier curves and study their algebraic and geometric properties. We show that the G-blossom provides the dual functionals for the G-Bernstein basis functions and we use this dual functional property to prove that G-Bernstein basis functions form a partition of unity and satisfy a Marsden identity. We also show that G-Bézier curves share several other properties with classical Bézier curves, including affine invariance, interpolation of end points, and recursive algorithms for evaluation and subdivision. We investigate the effect of the linear functions G on the shape of the corresponding G-Bézier curves, and we derive some necessary and sufficient conditions on the linear functions G which guarantee that the corresponding G-Bézier curves are of Pólya type and variation diminishing. Finally we prove that the control polygons generated by recursive subdivision converge to the original G-Bézier curve, and we derive the geometric rate of convergence of this algorithm.     

Asymptotics of instability zones of the Hill operator with a two term potential

Let γn denote the length of the nth zone of instability of the Hill operator Ly=−y^″−[4tαcos2x+2α²cos4x]y, where α≠0, and either both α, t are real, or both are pure imaginary numbers. For even n we prove: if t, n are fixed, then for α→0

     

Phototransistor noise model based on noise measurements on PNP PIN phototransistors

A noise model for phototransistors in open base configuration is presented. The model was developed from the noise measurements on four different phototransistor designs. The extracted current gains from the noise model were verified by measured current gains from Gummel measurements. Furthermore the current noise spectral density was modeled and compared with the noise measurements. A difference of less than 12 % in the current gain was achieved. In addition to the extracted current gains also the most dominant shot noise terms including their values of each phototransistor are extracted.     

Implicit standard Jacobi gives high relative accuracy

We prove that the Jacobi algorithm applied implicitly on a decomposition A = XDX ^T of the symmetric matrix A, where D is diagonal, and X is well conditioned, computes all eigenvalues of A to high relative accuracy. The relative error in every eigenvalue is bounded by O(ek(X)){O(\epsilon \kappa (X))} , where e{\epsilon} is the machine precision and k(X) o ||X||₂·||X^-1||₂{\kappa(X)\equiv\|X\|_2\cdot\|X^{-1}\|_2} is the spectral condition number of X. The eigenvectors are also computed accurately in the appropriate sense. We believe that this is the first algorithm to compute accurate eigenvalues of symmetric (indefinite) matrices that respects and preserves the symmetry of the problem and uses only orthogonal transformations.