Domination on Sets and in H p

In this paper we characterize the subsets E of the unit disk with the following property : ${\rm If}\ 0\ < \ p \ \leq \infty\ {\rm and}\ f \ \epsilon\ H^{p},\ 0 \ < \ ||f||_{p}\ <c \ < \infty $ then there exists $ {\cal G}\epsilon {H^{p}} $ such that $|{\cal G}(z)|\leq |f(z)|\ {\rm on}\ E\ {\rm and}\ ||{\cal G}||_{p}$ . A similar characterisation is given also for the case of the disk algebra.     

Necessary and sufficient conditions for the asymptotic distribution of the largest entry of a sample correlation matrix

Let {X _k,i ; i ≥ 1, k ≥ 1} be a double array of nondegenerate i.i.d. random variables and let {p _n ; n ≥ 1} be a sequence of positive integers such that n/p _n is bounded away from 0 and ∞. In this paper we give the necessary and sufficient conditions for the asymptotic distribution of the largest entry ${L_{n}={\rm max}_{1\leq i < j\leq p_{n}}|\hat{\rho}^{(n)}_{i,j}|}$ of the sample correlation matrix ${{\bf {\Gamma}}_{n}=(\hat{\rho}_{i,j}^{(n)})_{1\leq i,j\leq p_{n}}}$ where ${\hat{\rho}^{(n)}_{i,j}}$ denotes the Pearson correlation coefficient between (X _1,i , ..., X _n,i )′ and (X _1,j ,...,X _n,j )′. Write ${F(x)= \mathbb{P}(|X_{1,1}|\leq x), x\geq0}$ , ${W_{c,n}={\rm max}_{1\leq i < j\leq p_{n}}|\sum_{k=1}^{n}(X_{k,i}-c)(X_{k,j}-c)|}$ , and ${W_{n}=W_{0,n},n\geq1,c\in(-\infty,\infty)}$ . Under the assumption that ${\mathbb{E}|X_{1,1}|^{2+\delta} < \infty}$ for some δ > 0, we show that the following six statements are equivalent: $$ {\bf (i)} \quad \lim_{n \to \infty} n^{2}\int\limits_{(n \log n)^{1/4}}^{\infty}\left( F^{n-1}(x) - F^{n-1}\left(\frac{\sqrt{n \log n}}{x}\right) \right) dF(x) = 0,$$ $$ {\bf (ii)}\quad n \mathbb{P}\left ( \max_{1 \leq i < j \leq n}|X_{1,i}X_{1,j} | \geq \sqrt{n \log n}\right ) \to 0 \quad{\rm as}\,n \to \infty,$$ $$ {\bf (iii)}\quad \frac{W_{\mu, n}}{\sqrt {n \log n}}\stackrel{\mathbb{P}}{\rightarrow} 2\sigma^{2},$$ $$ {\bf (iv)}\quad \left ( \frac{n}{\log n}\right )^{1/2} L_{n} \stackrel{\mathbb{P}}{\rightarrow} 2,$$ $$ {\bf (v)}\quad \lim_{n \rightarrow \infty}\mathbb{P}\left (\frac{W_{\mu, n}^{2}}{n \sigma^{4}} - a_{n}\leq t \right ) = \exp \left \{ - \frac{1}{\sqrt{8\pi}} e^{-t/2}\right \}, - \infty < t < \infty,$$ $$ {\bf (vi)}\quad \lim_{n \rightarrow \infty}\mathbb{P}\left (n L_{n}^{2} - a_{n}\leq t \right ) = \exp \left \{ - \frac{1}{\sqrt{8 \pi}} e^{-t/2}\right \}, - \infty < t < \infty$$ where ${\mu=\mathbb{E}X_{1,1}, \sigma^{2}=\mathbb{E}(X_{1,1} - \mu)^{2}}$ , and a _n  = 4 log p _n ? log log p _n . The equivalences between (i), (ii), (iii), and (v) assume that only ${\mathbb{E}X_{1,1}^{2} < \infty}$ . Weak laws of large numbers for W _n and L _n , n ≥  1, are also established and these are of the form ${W_{n}/n^{\alpha}\stackrel{\mathbb{P}}{\rightarrow} 0}\,(\alpha > 1/2)$ and ${n^{1-\alpha}L_{n}\stackrel{\mathbb{P}}{\rightarrow} 0}\,(1/2 < \alpha \leq 1)$ , respectively. The current work thus provides weak limit analogues of the strong limit theorems of Li and Rosalsky as well as a necessary and sufficient condition for the asymptotic distribution of L _n obtained by Jiang. Some open problems are also posed.     

A convexity property and a new characterization of Euler’s gamma function

Our main results are:

1. Let α ≠ 0 be a real number. The function (Γ ? exp) ^α is convex on ${\mathbf{R}}$ if and only if $$\alpha \geq \max_{0<{t}<{x_0}}\Big(-\frac{1}{t\psi(t)} - \frac{\psi'(t)}{\psi(t)^2}\Big) = 0.0258... .$$ Here, x ₀ = 1.4616... denotes the only positive zero of ${\psi = \Gamma'/\Gamma}$ .

1. Assume that a function f: (0, ∞) → (0, ∞) is bounded from above on a set of positive Lebesgue measure (or on a set of the second category with the Baire property) and satisfies $$f(x+1) = x f(x) \quad{\rm for}\quad{x > 0}\quad{\rm and}\quad{f(1) = 1}.$$

If there are a number b and a sequence of positive real numbers (a _n ) ${(n \in \mathbf{N})}$ with ${{\rm lim}_{n\to\infty} a_n =0}$ such that for every n the function ${(f \circ {\rm exp})^{a_n}}$ is Jensen convex on (b, ∞), then f is the gamma function.     

Phase transitions of iterated Higman-style well-partial-orderings

We elaborate Weiermann-style phase transitions for well-partial-orderings (wpo) determined by iterated finite sequences under Higman-Friedman style embedding with Gordeev’s symmetric gap condition. For every d-times iterated wpo ${\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d}\right)}$ in question, d >? 1, we fix a natural extension of Peano Arithmetic, ${T \supseteq \sf{PA}}$ , that proves the corresponding second-order sentence ${\sf{WPO}\left({\rm S}{\textsc{eq}}^{d}, \trianglelefteq _{d}\right) }$ . Having this we consider the following parametrized first-order slow well-partial-ordering sentence ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d}, r\right):}$ $$\left( \forall K > 0 \right) \left( \exists M > 0\right) \left( \forall x_{0},\ldots ,x_{M}\in {\rm S}\text{\textsc{eq}}^{d}\right)$$ $$\left( \left( \forall i\leq M\right) \left( \left| x_{i}\right| < K + r \left\lceil \log _{d} \left( i+1\right) \right\rceil \right)\rightarrow \left( \exists i < j \leq M \right) \left(x_{i} \trianglelefteq _{d} x_{j}\right) \right)$$ for a natural additive Seq ^d -norm |·| and r ranging over EFA-provably computable positive reals, where EFA is an abbreviation for IΔ ₀?+?exp. We show that the following basic phase transition clauses hold with respect to ${T = \Pi_{1}^{0}\sf{CA}_{ < \varphi ^{_{\left( d-1\right) }} \left(0\right) }}$ and the threshold point1.

1. If r <? 1 then ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d},r \right) }$ is provable in T.

1. If ${r > 1}$ then ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{d}, \trianglelefteq _{d},r \right) }$ is not provable in T.

Moreover, by the well-known proof theoretic equivalences we can just as well replace T by PA or ACA ₀ and ${\Delta _{1}^{1}\sf{CA}}$ , if d =? 2 and d =? 3, respectively.In the limit case d → ∞ we replaceEFA-provably computable reals r by EFA-provably computable functions ${f: \mathbb{N} \rightarrow \mathbb{R}_{+}}$ and prove analogous theorems. (In the sequel we denote by ${\mathbb{R}_{+}}$ the set of EFA-provably computable positive reals). In the basic case T?=? PA we strengthen the basic phase transition result by adding the following static threshold clause

1. ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2}, 1\right)}$ is still provable in T = PA (actually in EFA).

Furthermore we prove the following dynamic threshold clauses which, loosely speaking are obtained by replacing the static threshold t by slowly growing functions 1 _α given by ${1_{\alpha }\left( i\right)\,{:=}\,1+\frac{1}{H_{\alpha }^{-1}\left(i\right) }, H_{\alpha}}$ being the familiar fast growing Hardy function and ${H_{\alpha }^{-1}\left( i\right)\,{:=}\,\rm min \left\{ j \mid H_{\alpha } \left ( j\right) \geq i \right\}}$ the corresponding slowly growing inversion.

1. If ${\alpha < \varepsilon _{0}}$ , then ${\sf{SWP}\left({\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2}, 1_{\alpha}\right)}$ is provable in T = PA.

1. ${\sf{SWP}\left( {\rm S}\text{\textsc{eq}}^{2}, \trianglelefteq _{2},1_{\varepsilon _{0}}\right)}$ is not provable in T = PA.

We conjecture that this pattern is characteristic for all ${T\supseteq \sf{PA}}$ under consideration and their proof-theoretical ordinals o (T ), instead of ${\varepsilon _{0}}$ .     

Two-step iterative algorithms for hierarchical fixed point problems and variational inequality problems

Let C be a nonempty closed convex subset of a real Hilbert space H. Let Q:C→C be a fixed contraction and S,T:C→C be two nonexpansive mappings such that Fix(T)≠?. Consider the following two-step iterative algorithm: $$\begin{array}{@{}rll}x_{n+1}&=&\alpha_{n}Qx_{n}+(1-\alpha_{n})Ty_{n},\\[1.5mm]y_{n}&=&\beta_{n}Sx_{n}+(1-\beta_{n})x_{n},\quad n\geq0.\end{array}$$ It is proven that under appropriate conditions, the above iterative sequence {x _n } converges strongly to $\tilde{x}\in \mathrm{Fix}(T)$ which solves some variational inequality depending on a given criterion S, namely: find $\tilde{x}\in H$ ; $0\in (I-S)\tilde{x}+N_{\mathrm{Fix}(T)}\tilde{x}$ , where N _Fix(T) denotes the normal cone to the set of fixed points of T.     

Some properties of a hypergeometric function which appear in an approximation problem

In this paper we consider properties and power expressions of the functions $f:(-1,1)\rightarrow \mathbb{R }$ and $f_L:(-1,1)\rightarrow \mathbb{R }$ , defined by $$\begin{aligned} f(x;\gamma )=\frac{1}{\pi }\int \limits _{-1}^1 \frac{(1+xt)^\gamma }{\sqrt{1-t^2}}\,\mathrm{d}t \quad \text{ and}\quad f_L(x;\gamma )=\frac{1}{\pi }\int \limits _{-1}^1 \frac{(1+xt)^\gamma \log (1+x t)}{\sqrt{1-t^2}}\,\mathrm{d}t, \end{aligned}$$ respectively, where $\gamma $ is a real parameter, as well as some properties of a two parametric real-valued function $D(\,\cdot \,;\alpha ,\beta ) :(-1,1) \rightarrow \mathbb{R }$ , defined by $$\begin{aligned} D(x;\alpha ,\beta )= f(x;\beta )f(x;-\alpha -1)- f(x;-\alpha )f(x;\beta -1),\quad \alpha ,\beta \in \mathbb{R }. \end{aligned}$$ The inequality of Turán type $$\begin{aligned} D(x;\alpha ,\beta )>0,\quad -1<x<1, \end{aligned}$$ for $\alpha +\beta >0$ is proved, as well as an opposite inequality if $\alpha +\beta <0$ . Finally, for the partial derivatives of $D(x;\alpha ,\beta )$ with respect to $\alpha $ or $\beta $ , respectively $A(x;\alpha ,\beta )$ and $B(x;\alpha ,\beta )$ , for which $A(x;\alpha ,\beta )=B(x;-\beta ,-\alpha )$ , some results are obtained. We mention also that some results of this paper have been successfully applied in various problems in the theory of polynomial approximation and some “truncated” quadrature formulas of Gaussian type with an exponential weight on the real semiaxis, especially in a computation of Mhaskar–Rahmanov–Saff numbers.     

On the existence of singular solutions of the stationary Navier–Stokes problem

We consider the steady Navier–Stokes equations in the punctured regions (?) Ω?=?Ω ₀ \ {o} (with {o} ∈ Ω ₀) and (??) $ \varOmega ={{\mathbb{R}}^2}\backslash \left( {{{\overline{\varOmega}}_0}\cup \left\{ o \right\}} \right) $ (with $ \left\{ o \right\}\notin {{\overline{\varOmega}}_0} $ ), where Ω ₀ is a simple connected Lipschitz bounded domain of $ {{\mathbb{R}}^2} $ . We regard o as a sink or a source in the fluid. Accordingly, we assign the flux $ \mathcal{F} $ through a small circumference surrounding o and a boundary datum a on Γ?=??Ω ₀ such that the total flux $ \mathcal{F}+\int\nolimits_{\varGamma } {\boldsymbol{a}\cdot \boldsymbol{n}} $ is zero in case (?). We prove that if $ \left| \mathcal{F} \right|<2\pi \nu $ and $ \left| \mathcal{F} \right|+\left| {\int\nolimits_{\varGamma } {\boldsymbol{a}\cdot \boldsymbol{n}} } \right|<2\pi \nu $ in (?) and (??), respectively, where ν is the kinematical viscosity, then the problem has a C ^∞ solution in Ω, which behaves at o like the gradient of the fundamental solution of the Laplace equation.     

The universality theorem for Hecke L-functions

We extend the universality theorem for Hecke L-functions attached to ray class characters from the previously known strip ${ \max \{\frac{1}{2}, 1-\frac{1}{d}\} < {\rm Re}\,s < 1}$ for ${d=\left[K:\mathbb{Q}\right]}$ to the maximal strip ${\frac{1}{2} < {\rm Re}\,s < 1}$ under an assumption of a weak version of the density hypothesis. As a corollary, we give a new proof of the universality theorem for the Dedekind zeta function ζ _K (s) in the case of ${K/\mathbb{Q}}$ finite abelian.     

Relations between the K-loop and the defect of an absolute plane

Let ${\cal P}$ be the point set of an absolute plane, let ${\cal {\tilde P}}$ be the set of all point reflections, let ?, resp. ?⁺, be the group of all, resp. of all proper, motions and let $$^\sim:{\cal P\times P\rightarrow \tilde P};\ \ \ (a,\ b)\mapsto\ \widetilde {a,\ b}$$ be the map where ${\widetilde {a,\ b}}$ denotes the uniquely determined point-reflection interchanging a and b. Then $$\delta\:\ {\cal P}^{3}\rightarrow {\cal M}^{+};\ \ \ (a,b,c)\mapsto \delta_{a;b,c}\:=\ {\tilde a}\ {\rm o}\ \widetilde {a,\ b}\ {\rm o}\ \widetilde {b,\ c}\ {\rm o}\ \widetilde {c,\ a}$$ is called the defect function, or shortly the defect. We show that δ_a;b,c is a rotation around the point a where the angle of δ_a;b,c is exactly the angle defect of the triangle (a, b, c) (cfr. 3.5). After fixing a point $o\ \in {\cal P}$ and setting $a+b\:=\widetilde {o,\ a}\ {\rm o}\ {\tilde o}\ (b),\ ({\cal P},+)$ becomes a K-loop and the so called precession function $$\delta_{a,b}\:=\ \big((a+b)^{+}\big)^{-1}\ {\rm o}\ a^{+}\ {\rm o}\ b^+$$ of the loop ( ${\cal P}, +)$ coincides with the defect of the triangle (o, a, ?b) (cfr. (4.4.1)), hence δ_a,b = δ_o;a,?b for all $a, b \in {\cal P}$ . With the order relation of the absolute plane we associate an orientation function $$\Omega\:\ \Delta\ \times \Delta \rightarrow \lbrace -1,+1\rbrace$$ defined on the pairs of triangles (cfr. (2.8)). If (a, b, c) ∈ Δ is a triangle and d a point of the line $\overline {b,\ c}$ ¹, then (cfr. (3.9.2)): $$\delta_{a;b,c}\ {\rm o}\ \delta_{a;c,d}=\delta_{a;b,d}$$ and moreover, if d is even a point of the open segment ]b, c[ then (cfr. (2.8.5)): $$\Omega(a,\ b,\ c;\ a,\ b,\ d)=\Omega(a,\ b,\ d;\ a,\ d,\ c)=+1.$$ Thus the angle defect of the triangle (a, b, c) is the sum of the angle defects of the triangles (a, b, d) and (a, d, c).     

Young measures generated by sequences in Morrey spaces

Let ${\Omega\subset\mathbb{R}^n}$ be open and bounded. For 1 ≤ p < ∞ and 0 ≤ λ < n, we give a characterization of Young measures generated by sequences of functions ${\{{\bf f}_j\}_{j=1}^\infty}$ uniformly bounded in the Morrey space ${L^{p,\lambda}(\Omega;\mathbb{R}^N)}$ with ${\{\left|{{\bf f}_j}\right|^p\}_{j=1}^\infty}$ equiintegrable. We then treat the case that each f _j = ? u _j for some ${{\bf u}_j\in W^{1,p}(\Omega;\mathbb{R}^N)}$ . As an application of our results, we consider the functional $${\bf u} \mapsto \int\limits_{\Omega}f({\bf x}, {\bf u}({\bf x}), {\bf {\nabla}}{\bf u}({\bf x})){\rm d}{\bf x},$$ and provide conditions that guarantee the existence of a minimizing sequence with gradients uniformly bounded in ${L^{p,\lambda}(\Omega;\mathbb{R}^{N\times n})}$ .     

Recurrence in β-expansion over formal Laurent series

In this paper, we study the quantitative recurrence and hitting sets of β-transformation T _β on the unit disk I of formal Laurent series field $$E_\phi:= \{x\in I: \|T_\beta^nx - x\| < \|\beta\|^{-\phi(n)}\,\,\,{\rm infinitely\,often}\}$$ and $$F_\phi:=\{x\in I: \|T_\beta^nx-x_0\|<\|\beta\|^{-\phi(n)}\,\,\,{\rm infinitely\,often}\},$$ where x ₀ is any fixed point in I and ${\phi}$ is any positive function defined on ${\mathbb{N}}$ with ${\phi(n)\to\infty}$ as n → ∞. We completely determine the Hausdorff dimensions of these sets: $$\dim_{\rm H} E_{\phi}=\dim_{\rm H}F_\phi=\frac{1}{1+\liminf\limits_{n\to\infty}\frac{\phi (n)}{n}}.$$      

A decomposition of the universal embedding space for the near polygon {{\mathbb H}_n}

Let ${{\mathbb H}_n, n \geq 1}$ , be the near 2n-gon defined on the 1-factors of the complete graph on 2n?+?2 vertices, and let e denote the absolutely universal embedding of ${{\mathbb H}_n}$ into PG(W), where W is a ${\frac{1}{n+2} \left(\begin{array}{c}2n+2 \\ n+1\end{array}\right)}$ -dimensional vector space over the field ${{\mathbb F}_2}$ with two elements. For every point z of ${{\mathbb H}_n}$ and every ${i \in {\mathbb N}}$ , let Δ _i (z) denote the set of points of ${{\mathbb H}_n}$ at distance i from z. We show that for every pair {x, y} of mutually opposite points of ${{\mathbb H}_n, W}$ can be written as a direct sum ${W_0 \oplus W_1 \oplus \cdots \oplus W_n}$ such that the following four properties hold for every ${i \in \{0,\ldots,n \}}$ : (1) ${\langle e(\Delta_i(x) \cap \Delta_{n-i}(y)) \rangle = {\rm PG}(W_i)}$ ; (2) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(x) \right) \right\rangle = {\rm PG}(W_0 \oplus W_1 \oplus \cdots \oplus W_i)}$ ; (3) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(y) \right) \right\rangle = {\rm PG}(W_{n-i}\oplus W_{n-i+1} \oplus \cdots \oplus W_n)}$ ; (4) ${\dim(W_i) = |\Delta_i(x) \cap \Delta_{n-i}(y)| = \left(\begin{array}{c}n \\ i\end{array}\right)^2 - \left(\begin{array}{c}n \\ i-1\end{array}\right) \cdot \left(\begin{array}{c}n \\ i+1\end{array}\right)}$ .     

Curvature estimates for submanifolds with prescribed Gauss image and mean curvature

We study that the n-graph defined by a smooth map ${f:\Omega\subset\mathbb R^{n}\to \mathbb R^{m}, m\ge 2,}$ in ${\mathbb R^{m+n}}$ of the prescribed mean curvature and the Gauss image. Under the condition $$\Delta_f=\left[\text{det}\left(\delta_{ij}+\sum_\alpha{\frac {\partial {f^\alpha}}{\partial {x^i}}}{\frac {\partial {f^\alpha}}{\partial {x^j}}}\right)\right]^{\frac{1}{2}} < 2,$$ we derive the interior curvature estimates $$\sup_{D_R(x)}|B|^2\le{\frac{C}{R^2}}$$ when 2 ≤ n ≤ 5 with constant C depending on the given geometric data. If there is no dimension limitation we obtain $$\sup_{D_R(x)}|B|^2\le CR^{-a}\sup_{D_{2R}(x)}(2-\Delta_f)^{-\left({\frac{3}{2}}+{\frac{1}{s}}\right)},\quad s=\min(m, n)$$ with a < 1. If the image under the Gauss map is contained in a geodesic ball of the radius ${{\frac{\sqrt{2}}{4}}\pi}$ in G _n,m we also derive corresponding estimates.     

Eigenvalue comparisons for second order difference equations with periodic and antiperiodic boundary conditions

We consider a class of boundary value problems of the second order difference equation $$\Delta(r_{i-1}\Delta y_{i-1})-b_{i}y_{i}+\lambda a_{i}y_{i}=0,\quad 1\le i\le n,\ y_{0}=\alpha y_{n},\ y_{n+1}=\alpha y_{1}.$$ The class of problems considered includes those with antiperiodic, Dirichlet, and periodic boundary conditions. We focus on the structure of eigenvalues of this class of problems and comparisons of all eigenvalues as the coefficients {a _i } _i=1 ⁿ ,{b _i } _i=1 ⁿ , and {r _i } _i=0 ⁿ change.     

Full regularity of a free boundary problem with two phases

Let ?? be a bounded domain in ${\mathbb{R}^{n}, n\geq2}$ . We use ${\mathcal{M}_{\Omega}}$ to denote the collection of all pairs of (A, u) such that ${A\subset\Omega}$ is a set of finite perimeter and ${u\in H^{1}\left( \Omega\right)}$ satisfies $$u\left( x\right) =0\quad\text{a.e.}x\in A.$$ We consider the energy functional $$E_{\Omega}\left( A,u\right) =\int\limits_{\Omega}\left\vert\triangledown u\right\vert ^{2}+P_{\Omega}\left( A\right)$$ defined on ${\mathcal{M}_{\Omega}}$ , where P _??(A) denotes the perimeter of A inside ??. Let ${\left( A,u\right)\in\mathcal{M}_{\Omega}}$ be a minimizer with volume constraint. Our main result is that when n????7, u is locally Lipschitz and the free boundary ?A is analytic in ??.     

The formal translation equation for iteration groups of type II

We investigate the translation equation $$F(s+t, x) = F(s, F(t, x)),\quad \quad s,t\in{\mathbb{C}},\qquad\qquad\qquad\qquad({\rm T})$$ in ${\mathbb{C}\left[\kern-0.15em\left[{x}\right]\kern-0.15em\right]}$ , the ring of formal power series over ${\mathbb{C}}$ . Here we restrict ourselves to iteration groups of type II, i.e. to solutions of (T) of the form ${F(s, x) \equiv x + c_k(s)x^k {\rm mod} x^{k + 1}}$ , where k ≥ 2 and c _k ≠ 0 is necessarily an additive function. It is easy to prove that the coefficient functions c _n (s) of $$F(s, x) = x + \sum_{n \ge q k}c_n(s)x^n$$ are polynomials in c _k (s). It is possible to replace this additive function c _k by an indeterminate. In this way we obtain a formal version of the translation equation in the ring ${(\mathbb{C}[y])\left[\kern-0.15em\left[{x}\right]\kern-0.15em\right]}$ . We solve this equation in a completely algebraic way, by deriving formal differential equations or an Aczél–Jabotinsky type equation. This way it is possible to get the structure of the coefficients in great detail which are now polynomials. We prove the universal character (depending on certain parameters, the coefficients of the infinitesimal generator H of an iteration group of type II) of these polynomials. Rewriting the solutions G(y, x) of the formal translation equation in the form ${\sum_{n\geq 0}\phi_n(x)y^n}$ as elements of ${(\mathbb{C}\left[\kern-0.15em\left[{x}\right]\kern-0.15em\right])\left[\kern-0.15em\left[{y}\right]\kern-0.15em\right]}$ , we obtain explicit formulas for ${\phi_n}$ in terms of the derivatives H ^(j)(x) of the generator ${H}$ and also a representation of ${G(y, x)}$ as a Lie–Gröbner series. Eventually, we deduce the canonical form (with respect to conjugation) of the infinitesimal generator ${H}$ as x ^k + hx ^2k-1 and find expansions of the solutions ${G(y, x) = \sum_{r\geq 0} G_r(y, x)h^r}$ of the above mentioned differential equations in powers of the parameter h.     

Limiting Behavior of High Order Correlations for Simple Random Sampling

For ${N = 1, 2,\ldots,}$ let S _N be a simple random sample of size n = n _N from a population A _N of size N, where ${0 \leq n \leq N}$ . Then with f _N =  n/N, the sampling fraction, and 1 _A the inclusion indicator that ${A \in S_N}$ , for any ${H \subset A_N}$ of size ${k \geq 0}$ , the high order correlations $${\rm Corr}(k) = E \big(\mathop{\Pi}\limits_{A \in H} ({\bf 1}_A - f_N )\big)$$ depend only on k, and if the sampling fraction ${f_N \rightarrow f}$ as ${N \rightarrow \infty}$ , then $$N^{k/2}{\rm Corr}(k) \rightarrow [f (f - 1)]^{k/2}EZ^{k}, k \,{\rm even}$$ and $$N^{(k+1)/2}{\rm Corr}(k) \rightarrow [f (f - 1)]^{(k-1)/2}(2f - 1)\frac{1}{3}(k - 1)EZ^{k+1}, k \,{\rm odd}$$ where Z is a standard normal random variable. This proves a conjecture given in [2].