Fulkersonʼs Conjecture and Loupekine snarks

Fulkersonʼs Conjecture says that every bridgeless cubic graph has six perfect matchings such that each edge belongs to exactly two of them. In 1976, F. Loupekine created a method for constructing new snarks from already known ones. We consider an infinite family of snarks built with Loupekineʼs method, and verify Fulkersonʼs Conjecture for this family.     

Quadratic Hermite-Padé approximation to the exponential function

Approximation to exp of the form wherep _m,q _m, andr _m are polynomials of degree at mostm andp _m has lead coefficient 1 is considered. Exact asymptotics and explicit formulas are obtained for the sequences {? _m}, {p _m}, {q _m}, and {r _m}. It is observed that the above sequences all satisfy the simple four-term recursion: $$\begin{array}{*{20}c} {T_{m + 3} = \frac{1}{{3m + 4}}[( - 6m - 14)z^3 T_m } \\ { + (9m + 15)(z^2 + (3m + 4)(3m + 7))T_{m + 1} + 3zT_{m + 2} ].} \\ \end{array} $$ It is also observed that these generalized Padé-type approximations can be used to asymptotically minimize expressions of the above form on the unit disk.     

Zhang’s Conjecture and the Effective Bogomolov Conjecture over function fields

We prove the Effective Bogomolov Conjecture, and so the Bogomolov Conjecture, over a function field of characteristic 0 by proving Zhang’s Conjecture about certain invariants of metrized graphs. In the function field case, these conjectures were previously known to be true only for curves of good reduction, for curves of genus at most 4 and a few other special cases. We also either verify or improve the previous results. We relate the invariants involved in Zhang’s Conjecture to the tau constant of metrized graphs. Then we use and extend our previous results on the tau constant. By proving another Conjecture of Zhang, we obtain a new proof of the slope inequality for Faltings heights on moduli space of curves.     

Fischer determinantal inequalities and Highamʼs Conjecture

We obtain a Fischer type determinantal inequality for matrices with given angular numerical range. We discuss the growth factor for Gaussian elimination for linear systems in which the coefficient matrix has this form and give a proof of Higham?s Conjecture.     

Nevanlinna theory and Diophantine approximation

We discuss the analogue of the Nevanlinna theory and the theory of Diophan-tine approximation, focussing on the second main theorem and abc-conjecture.     

The period function of generalized Loudʼs centers

In this paper a three parameter family of planar differential systems with homogeneous nonlinearities of arbitrary odd degree is studied. This family is an extension to higher degree of Loud?s systems. The origin is a nondegenerate center for all values of the parameter and we are interested in the qualitative properties of its period function. We study the bifurcation diagram of this function focusing our attention on the bifurcations occurring at the polycycle that bounds the period annulus of the center. Moreover we determine some regions in the parameter space for which the corresponding period function is monotonous or it has at least one critical period, giving also its character (maximum or minimum). Finally we propose a complete conjectural bifurcation diagram of the period function of these generalized Loud?s centers.     

Approximation of Sondow’s generalized-Euler-constant function on the interval [−1, 1]

Using the Euler-Maclaurin (Boole/Hermite) summation formula, the generalized-Euler-Sondow-constant function γ(z),

$ \gamma(z):=\sum_{k=1}^{\infty}z^{k-1}\left(\frac{1}{k}-\ln\frac{k+1}{k}\right) \qquad (-1\le z\le 1),$

where \({\gamma(-1)=\ln\frac{4}{\pi}}\) and γ(1) is the Euler-Mascheroni constant, is estimated accurately.

     

Extensions of the disc algebra and of Mergelyanʼs theorem

We investigate the uniform limits of the set of polynomials on the closed unit disc $\overline{D}$ with respect to the chordal metric χ. More generally, we examine analogous questions replacing $\mathbb{C}\cup \{\infty \}$ by other metrizable compactifications of $\mathbb{C}$.     

The Fourier–Stieltjes transform of Minkowskiʼs ?(x) function and an affirmative answer to Salemʼs problem

By using structural and asymptotic properties of the Kontorovich–Lebedev transform associated with Minkowski?s question mark function, we give an affirmative answer to the question posed by R. Salem (Trans. Amer. Math. Soc. 53 (3) (1943) 439) whether its Fourier–Stieltjes transform vanishes at infinity.     

A quicker continued fraction approximation of the gamma function related to the Windschitl’s formula

In this paper, based on the Windschitl’s formula, a new continued fraction approximation and some inequalities for the gamma function are established. Finally, for demonstrating the superiority of our new series over the classical ones, some numerical computations are given.     

Sets of exact ‘logarithmic’ order in the theory of Diophantine approximation

For each real number , let denote the set of real numbers with exact order . A theorem of Güting states that for the Hausdorff dimension of is equal to . In this note we introduce the notion of exact t–logarithmic order which refines the usual definition of exact order. Our main result for the associated refined sets generalizes Güting's result to linear forms and moreover determines the Hausdorff measure at the critical exponent. In fact, the sets are shown to satisfy delicate zero-infinity laws with respect to Lebesgue and Hausdorff measures. These laws are reminiscent of those satisfied by the classical set of well approximable real numbers, for example as demonstrated by Khintchine's theorem. Received: 12 December 2000 / Published online: 25 June 2001