Volume 3, Number 6 - March 1923

Volume 3, Number 6 - March 1923. 38 pages including covers and advertisements. Contents Eldy, Francis, The Silver Crown Keliher, J. F., The Indubitable Thomas Gibbon, Charles A., Try It Dwyer, Francis L., Unusual Boppell, Leo J., A Real Short Story K., J. F., Spring of Life Lynch, James H., 135 Bedside Eldy, Francis, Silver Plated Said the Walrus to the Carpenter K., J. F., Filio Dominici Editorial K., J. F., Youth Mitchell, J., College Chronicle Simpson, V. F., Eternal Promise Olivier, L., Exchang

Exponential Formulas for the Jacobians and Jacobian Matrices of Analytic Maps

Let $F=(F_1, F_2, ... F_n)$ be an $n$-tuple of formal power series in $n$ variables of the form $F(z)=z+ O(|z|^2)$. It is known that there exists a unique formal differential operator A=\sum_{i=1}^n a_i(z)\frac {\p}{\p z_i} such that $F(z)=exp (A)z$ as formal series. In this article, we show the Jacobian ${\cal J}(F)$ and the Jacobian matrix $J(F)$ of $F$ can also be given by some exponential formulas. Namely, ${\cal J}(F)=\exp (A+\triangledown A)\cdot 1$, where \triangledown A(z)= \sum_{i=1}^n \frac {\p a_i}{\p z_i}(z), and $J(F)=\exp(A+R_{Ja})\cdot I_{n\times n}$, where $I_{n\times n}$ is the identity matrix and $R_{Ja}$ is the multiplication operator by $Ja$ for the right. As an immediate consequence, we get an elementary proof for the known result that ${\cal J}(F)\equiv 1$ if and only if $\triangledown A=0$. Some consequences and applications of the exponential formulas as well as their relations with the well known Jacobian Conjecture are also discussed.Comment: Latex, 17 page

Semiconjugacies, pinched Cantor bouquets and hyperbolic orbifolds

Let f be a transcendental entire map that is subhyperbolic, i.e., the intersection of the Fatou set F(f) and the postsingular set P(f) is compact and the intersection of the Julia set J(f) and P(f) is finite. Assume that no asymptotic value of f belongs to J(f) and that the local degree of f at all points in J(f) is bounded by some finite constant. We prove that there is a hyperbolic map g (of the form g(z)=f(bz) for some complex number b) with connected Fatou set such that f and g are semiconjugate on their Julia sets. Furthermore, we show that this semiconjugacy is a conjugacy when restricted to the escaping set I(g) of g. In the case where f can be written as a finite composition of maps of finite order, our theorem, together with recent results on Julia sets of hyperbolic maps, implies that J(f) is a pinched Cantor bouquet, consisting of dynamic rays and their endpoints. Our result also seems to give the first complete description of topological dynamics of an entire transcendental map whose Julia set is the whole complex plane.Comment: 32 pages, 3 figure

A note on the strong law of large numbers

identically distributed (i.i.d.) random variables. Let Sn-t^Xk (fl » 1, 2, • • •)• J f e- 1 A long standing problem in probability theory has been to find neces