Abstract:
Maternal diabetes induces fetal alterations, resulting in lasting consequences for the glucose tolerance of the offspring over several generations. In our experimental rat model, circulating prolactin, oestradiol, progesterone and corticosterone levels, known to influence insulin secretion and action, are determined in plasma of female adult offspring of mildly and severely diabetic mothers. Prolactin and progesterone levels are equally low in both groups as compared to controls, stressing the involvement of the CNS in the transgeneration effect; oestradiol and corticosterone levels are normal. No correlation is found between these hormonal alterations and the known differences in glucose tolerance.

Abstract:
Orthogonal polynomials on the real line always satisfy a three-term recurrence relation. The recurrence coefficients determine a tridiagonal semi-infinite matrix (Jacobi matrix) which uniquely characterizes the orthogonal polynomials. We investigate new orthogonal polynomials by adding to the Jacobi matrix $r$ new rows and columns, so that the original Jacobi matrix is shifted downward. The $r$ new rows and columns contain $2r$ new parameters and the newly obtained orthogonal polynomials thus correspond to an upward extension of the Jacobi matrix. We give an explicit expression of the new orthogonal polynomials in terms of the original orthogonal polynomials, their associated polynomials and the $2r$ new parameters, and we give a fourth order differential equation for these new polynomials when the original orthogonal polynomials are classical. Furthermore we show how the orthogonalizing measure for these new orthogonal polynomials can be obtained and work out the details for a one-parameter family of Jacobi polynomials for which the associated polynomials are again Jacobi polynomials.

Abstract:
We show that the alternative discrete Painlev\'e I equation (alt-dP$_{\rm I}$) has a unique solution which remains positive for all $n \geq 0$. Furthermore, we identify this positive solution in terms of a special solution of the second Painlev\'e equation (P$_{\rm II}$) involving the Airy function $\mathop{\rm Ai}(t)$. The special-function solutions of P$_{\rm II}$ involving only the Airy function $\mathop{\rm Ai}(t)$ therefore have the property that they remain positive for all $n\geq 0$ and all $t \geq 0$, which is a new characterization of these special solutions of P$_{\rm II}$ and alt-dP$_{\rm I}$.

Abstract:
We investigate the asymptotic behavior of the polynomials p, q, r of degrees n in type I Hermite-Pade approximation to the exponential function, defined by p(z)e^{-z}+q(z)+r(z)e^{z} = O(z^{3n+2}) as z -> 0. These polynomials are characterized by a Riemann-Hilbert problem for a 3x3 matrix valued function. We use the Deift-Zhou steepest descent method for Riemann-Hilbert problems to obtain strong uniform asymptotics for the scaled polynomials p(3nz), q(3nz), and r(3nz) in every domain in the complex plane. An important role is played by a three-sheeted Riemann surface and certain measures and functions derived from it. Our work complements recent results of Herbert Stahl.

Abstract:
Quantum continuous variables are being explored as an alternative means to implement quantum key distribution, which is usually based on single photon counting. The former approach is potentially advantageous because it should enable higher key distribution rates. Here we propose and experimentally demonstrate a quantum key distribution protocol based on the transmission of gaussian-modulated coherent states (consisting of laser pulses containing a few hundred photons) and shot-noise-limited homodyne detection; squeezed or entangled beams are not required. Complete secret key extraction is achieved using a reverse reconciliation technique followed by privacy amplification. The reverse reconciliation technique is in principle secure for any value of the line transmission, against gaussian individual attacks based on entanglement and quantum memories. Our table-top experiment yields a net key transmission rate of about 1.7 megabits per second for a loss-free line, and 75 kilobits per second for a line with losses of 3.1 dB. We anticipate that the scheme should remain effective for lines with higher losses, particularly because the present limitations are essentially technical, so that significant margin for improvement is available on both the hardware and software.

Abstract:
We obtain strong and uniform asymptotics in every domain of the complex plane for the scaled polynomials $a (3nz)$, $b (3nz)$, and $c (3nz)$ where $a$, $b$, and $c$ are the type II Hermite-Pad\'e approximants to the exponential function of respective degrees $2n+2$, $2n$ and $2n$, defined by $a (z)e^{-z}-b (z)=\O (z^{3n+2})$ and $a (z)e^{z}-c (z)={\O}(z^{3n+2})$ as $z\to 0$. Our analysis relies on a characterization of these polynomials in terms of a $3\times 3$ matrix Riemann-Hilbert problem which, as a consequence of the famous Mahler relations, corresponds by a simple transformation to a similar Riemann-Hilbert problem for type I Hermite-Pad\'e approximants. Due to this relation, the study that was performed in previous work, based on the Deift-Zhou steepest descent method for Riemann-Hilbert problems, can be reused to establish our present results.

Abstract:
We give a short introduction to Padé approximation (rational approximation to a function with close contact at one point) and to Hermite-Padé approximation (simultaneous rational approximation to several functions with close contact at one point) and show how orthogonality plays a crucial role. We give some insight into how logarithmic potential theory helps in describing the asymptotic behavior and the convergence properties of Padé and Hermite-Padé approximation.

Abstract:
We give a short introduction to Pade approximation (rational approximation to a function with close contact at one point) and to Hermite-Pade approximation (simultaneous rational approximation to several functions with close contact at one point) and show how orthogonality plays a crucial role. We give some insight into how logarithmic potential theory helps in describing the asymptotic behavior and the convergence properties of Pade and Hermite-Pade approximation.

Abstract:
We give the asymptotic behavior of the ratio of two neighboring multiple orthogonal polynomials under the condition that the recurrence coefficients in the nearest neighbor recurrence relations converge.